poimirlem15 - Mathbox for Brendan Leahy

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Theorem poimirlem15 33554

Description: Lemma for poimir 33572, that the face in poimirlem22 33561 is a face. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
poimirlem22.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem15.3	⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))

**Assertion**
Ref	Expression
poimirlem15	⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ 𝑆)

Distinct variable groups: 𝑓,𝑗,𝑡,𝑦 𝜑,𝑗,𝑦 𝑗,𝐹,𝑦 𝑗,𝑁,𝑦 𝑇,𝑗,𝑦 𝜑,𝑡 𝑓,𝐾,𝑗,𝑡 𝑓,𝑁,𝑡 𝑇,𝑓 𝑓,𝐹,𝑡 𝑡,𝑇 𝑆,𝑗,𝑡,𝑦 Allowed substitution hints: 𝜑(𝑓) 𝑆(𝑓) 𝐾(𝑦)

**Proof of Theorem poimirlem15**
Step	Hyp	Ref	Expression
1		poimirlem22.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		elrabi 3391	. . . . . . 7 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3		poimirlem22.s	. . . . . . 7 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4	2, 3	eleq2s 2748	. . . . . 6 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6		xp1st 7242	. . . . 5 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
7		xp1st 7242	. . . . 5 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
8	5, 6, 7	3syl 18	. . . 4 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
9		xp2nd 7243	. . . . . . . 8 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10	5, 6, 9	3syl 18	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
11		fvex 6239	. . . . . . . 8 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
12		f1oeq1 6165	. . . . . . . 8 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
13	11, 12	elab 3382	. . . . . . 7 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14	10, 13	sylib 208	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
15		poimirlem15.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
16		elfznn 12408	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
18	17	nnred 11073	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
19	18	ltp1d 10992	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
20	18, 19	ltned 10211	. . . . . . . . . 10 ⊢ (𝜑 → (2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1))
21	20	necomd 2878	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇))
22		fvex 6239	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑇) ∈ V
23		ovex 6718	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑇) + 1) ∈ V
24		f1oprg 6219	. . . . . . . . . . 11 ⊢ ((((2^nd ‘𝑇) ∈ V ∧ ((2^nd ‘𝑇) + 1) ∈ V) ∧ (((2^nd ‘𝑇) + 1) ∈ V ∧ (2^nd ‘𝑇) ∈ V)) → (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇)) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}))
25	22, 23, 23, 22, 24	mp4an 709	. . . . . . . . . 10 ⊢ (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇)) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
26	20, 21, 25	syl2anc 694	. . . . . . . . 9 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
27		prcom 4299	. . . . . . . . . 10 ⊢ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}
28		f1oeq3 6167	. . . . . . . . . 10 ⊢ ({((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
29	27, 28	ax-mp 5	. . . . . . . . 9 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
30	26, 29	sylib 208	. . . . . . . 8 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
31		f1oi 6212	. . . . . . . 8 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
32		disjdif 4073	. . . . . . . . 9 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅
33		f1oun 6194	. . . . . . . . 9 ⊢ ((({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ∧ (({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅ ∧ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))–1-1-onto→({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
34	32, 32, 33	mpanr12 721	. . . . . . . 8 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))–1-1-onto→({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
35	30, 31, 34	sylancl 695	. . . . . . 7 ⊢ (𝜑 → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))–1-1-onto→({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
36		poimir.0	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ)
37	36	nncnd 11074	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℂ)
38		npcan1 10493	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
39	37, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
40	36	nnzd 11519	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
41		peano2zm 11458	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
42	40, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
43		uzid 11740	. . . . . . . . . . . . . 14 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
44		peano2uz 11779	. . . . . . . . . . . . . 14 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
45	42, 43, 44	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
46	39, 45	eqeltrrd 2731	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
47		fzss2 12419	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
48	46, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
49	48, 15	sseldd 3637	. . . . . . . . . 10 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...𝑁))
50	17	peano2nnd 11075	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℕ)
51	42	zred 11520	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
52	36	nnred 11073	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
53		elfzle2 12383	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ≤ (𝑁 − 1))
54	15, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘𝑇) ≤ (𝑁 − 1))
55	52	ltm1d 10994	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
56	18, 51, 52, 54, 55	lelttrd 10233	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ‘𝑇) < 𝑁)
57	17	nnzd 11519	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℤ)
58		zltp1le 11465	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑇) < 𝑁 ↔ ((2^nd ‘𝑇) + 1) ≤ 𝑁))
59	57, 40, 58	syl2anc 694	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘𝑇) < 𝑁 ↔ ((2^nd ‘𝑇) + 1) ≤ 𝑁))
60	56, 59	mpbid 222	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ≤ 𝑁)
61		fznn 12446	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (((2^nd ‘𝑇) + 1) ∈ (1...𝑁) ↔ (((2^nd ‘𝑇) + 1) ∈ ℕ ∧ ((2^nd ‘𝑇) + 1) ≤ 𝑁)))
62	40, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) ∈ (1...𝑁) ↔ (((2^nd ‘𝑇) + 1) ∈ ℕ ∧ ((2^nd ‘𝑇) + 1) ≤ 𝑁)))
63	50, 60, 62	mpbir2and 977	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
64		prssi 4385	. . . . . . . . . 10 ⊢ (((2^nd ‘𝑇) ∈ (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁)) → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁))
65	49, 63, 64	syl2anc 694	. . . . . . . . 9 ⊢ (𝜑 → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁))
66		undif 4082	. . . . . . . . 9 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁))
67	65, 66	sylib 208	. . . . . . . 8 ⊢ (𝜑 → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁))
68		f1oeq23 6168	. . . . . . . 8 ⊢ ((({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁) ∧ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))–1-1-onto→({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ↔ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)))
69	67, 67, 68	syl2anc 694	. . . . . . 7 ⊢ (𝜑 → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))–1-1-onto→({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ↔ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)))
70	35, 69	mpbid 222	. . . . . 6 ⊢ (𝜑 → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁))
71		f1oco 6197	. . . . . 6 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
72	14, 70, 71	syl2anc 694	. . . . 5 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
73		prex 4939	. . . . . . . 8 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∈ V
74		ovex 6718	. . . . . . . . 9 ⊢ (1...𝑁) ∈ V
75		difexg 4841	. . . . . . . . 9 ⊢ ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ V)
76		resiexg 7144	. . . . . . . . 9 ⊢ (((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ V → ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ∈ V)
77	74, 75, 76	mp2b 10	. . . . . . . 8 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ∈ V
78	73, 77	unex 6998	. . . . . . 7 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) ∈ V
79	11, 78	coex 7160	. . . . . 6 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) ∈ V
80		f1oeq1 6165	. . . . . 6 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)))
81	79, 80	elab 3382	. . . . 5 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
82	72, 81	sylibr 224	. . . 4 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
83		opelxpi 5182	. . . 4 ⊢ (((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)) ∧ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
84	8, 82, 83	syl2anc 694	. . 3 ⊢ (𝜑 → ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85		fz1ssfz0 12474	. . . . 5 ⊢ (1...𝑁) ⊆ (0...𝑁)
86	48, 85	syl6ss 3648	. . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁))
87	86, 15	sseldd 3637	. . 3 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (0...𝑁))
88		opelxpi 5182	. . 3 ⊢ ((⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (2^nd ‘𝑇) ∈ (0...𝑁)) → ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89	84, 87, 88	syl2anc 694	. 2 ⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
90		fveq2 6229	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
91	90	breq2d 4697	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
92	91	ifbid 4141	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
93	92	csbeq1d 3573	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
94		fveq2 6229	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (1^st ‘𝑡) = (1^st ‘𝑇))
95	94	fveq2d 6233	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
96	94	fveq2d 6233	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
97	96	imaeq1d 5500	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
98	97	xpeq1d 5172	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
99	96	imaeq1d 5500	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
100	99	xpeq1d 5172	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
101	98, 100	uneq12d 3801	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
102	95, 101	oveq12d 6708	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
103	102	csbeq2dv 4025	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
104	93, 103	eqtrd 2685	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
105	104	mpteq2dv 4778	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
106	105	eqeq2d 2661	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
107	106, 3	elrab2 3399	. . . . 5 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
108	107	simprbi 479	. . . 4 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
109	1, 108	syl 17	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
110		imaco 5678	. . . . . . . . . 10 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) = ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...𝑦)))
111		f1ofn 6176	. . . . . . . . . . . . . . . 16 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
112	26, 111	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
113	112	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
114		incom 3838	. . . . . . . . . . . . . . 15 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
115		elfznn0 12471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
116	115	nn0red 11390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
117		ltnle 10155	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ ∧ (2^nd ‘𝑇) ∈ ℝ) → (𝑦 < (2^nd ‘𝑇) ↔ ¬ (2^nd ‘𝑇) ≤ 𝑦))
118	116, 18, 117	syl2anr 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^nd ‘𝑇) ↔ ¬ (2^nd ‘𝑇) ≤ 𝑦))
119	118	biimpa 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ¬ (2^nd ‘𝑇) ≤ 𝑦)
120		elfzle2 12383	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑇) ∈ (1...𝑦) → (2^nd ‘𝑇) ≤ 𝑦)
121	119, 120	nsyl 135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ¬ (2^nd ‘𝑇) ∈ (1...𝑦))
122		disjsn 4278	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑦) ∩ {(2^nd ‘𝑇)}) = ∅ ↔ ¬ (2^nd ‘𝑇) ∈ (1...𝑦))
123	121, 122	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((1...𝑦) ∩ {(2^nd ‘𝑇)}) = ∅)
124	116	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → 𝑦 ∈ ℝ)
125	18	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ ℝ)
126	50	nnred 11073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℝ)
127	126	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘𝑇) + 1) ∈ ℝ)
128		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → 𝑦 < (2^nd ‘𝑇))
129	19	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
130	124, 125, 127, 128, 129	lttrd 10236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → 𝑦 < ((2^nd ‘𝑇) + 1))
131		ltnle 10155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ ((2^nd ‘𝑇) + 1) ∈ ℝ) → (𝑦 < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ 𝑦))
132	116, 126, 131	syl2anr 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ 𝑦))
133	132	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (𝑦 < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ 𝑦))
134	130, 133	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ¬ ((2^nd ‘𝑇) + 1) ≤ 𝑦)
135		elfzle2 12383	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...𝑦) → ((2^nd ‘𝑇) + 1) ≤ 𝑦)
136	134, 135	nsyl 135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ¬ ((2^nd ‘𝑇) + 1) ∈ (1...𝑦))
137		disjsn 4278	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑦) ∩ {((2^nd ‘𝑇) + 1)}) = ∅ ↔ ¬ ((2^nd ‘𝑇) + 1) ∈ (1...𝑦))
138	136, 137	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((1...𝑦) ∩ {((2^nd ‘𝑇) + 1)}) = ∅)
139	123, 138	uneq12d 3801	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((1...𝑦) ∩ {(2^nd ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^nd ‘𝑇) + 1)})) = (∅ ∪ ∅))
140		df-pr 4213	. . . . . . . . . . . . . . . . . 18 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
141	140	ineq2i 3844	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...𝑦) ∩ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)}))
142		indi 3906	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑦) ∩ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})) = (((1...𝑦) ∩ {(2^nd ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^nd ‘𝑇) + 1)}))
143	141, 142	eqtr2i 2674	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑦) ∩ {(2^nd ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^nd ‘𝑇) + 1)})) = ((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
144		un0 4000	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ ∅) = ∅
145	139, 143, 144	3eqtr3g 2708	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅)
146	114, 145	syl5eq 2697	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (1...𝑦)) = ∅)
147		fnimadisj 6050	. . . . . . . . . . . . . 14 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (1...𝑦)) = ∅) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...𝑦)) = ∅)
148	113, 146, 147	syl2anc 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...𝑦)) = ∅)
149	39	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
150		elfzuz3 12377	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
151		peano2uz 11779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
152	150, 151	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
153	152	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
154	149, 153	eqeltrrd 2731	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝑦))
155		fzss2 12419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝑦) → (1...𝑦) ⊆ (1...𝑁))
156		reldisj 4053	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑦) ⊆ (1...𝑁) → (((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
157	154, 155, 156	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
158	157	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((1...𝑦) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
159	145, 158	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
160		resiima 5515	. . . . . . . . . . . . . 14 ⊢ ((1...𝑦) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
161	159, 160	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
162	148, 161	uneq12d 3801	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...𝑦))) = (∅ ∪ (1...𝑦)))
163		imaundir 5581	. . . . . . . . . . . 12 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...𝑦)) = (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...𝑦)))
164		uncom 3790	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (1...𝑦)) = ((1...𝑦) ∪ ∅)
165		un0 4000	. . . . . . . . . . . . 13 ⊢ ((1...𝑦) ∪ ∅) = (1...𝑦)
166	164, 165	eqtr2i 2674	. . . . . . . . . . . 12 ⊢ (1...𝑦) = (∅ ∪ (1...𝑦))
167	162, 163, 166	3eqtr4g 2710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...𝑦)) = (1...𝑦))
168	167	imaeq2d 5501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...𝑦))) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)))
169	110, 168	syl5eq 2697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)))
170	169	xpeq1d 5172	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}))
171		imaco 5678	. . . . . . . . . 10 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)))
172		imaundir 5581	. . . . . . . . . . . . 13 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)))
173		imassrn 5512	. . . . . . . . . . . . . . . . 17 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}
174	173	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
175		fnima 6048	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
176	26, 111, 175	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
177	176	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
178		elfzelz 12380	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
179		zltp1le 11465	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℤ ∧ (2^nd ‘𝑇) ∈ ℤ) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑦 + 1) ≤ (2^nd ‘𝑇)))
180	178, 57, 179	syl2anr 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑦 + 1) ≤ (2^nd ‘𝑇)))
181	180	biimpa 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (𝑦 + 1) ≤ (2^nd ‘𝑇))
182	18, 52, 56	ltled 10223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (2^nd ‘𝑇) ≤ 𝑁)
183	182	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (2^nd ‘𝑇) ≤ 𝑁)
184	57	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) ∈ ℤ)
185		nn0p1nn 11370	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
186	115, 185	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
187	186	nnzd 11519	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
188	187	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ)
189	40	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ)
190		elfz 12370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑇) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^nd ‘𝑇) ∧ (2^nd ‘𝑇) ≤ 𝑁)))
191	184, 188, 189, 190	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^nd ‘𝑇) ∧ (2^nd ‘𝑇) ≤ 𝑁)))
192	191	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^nd ‘𝑇) ∧ (2^nd ‘𝑇) ≤ 𝑁)))
193	181, 183, 192	mpbir2and 977	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ ((𝑦 + 1)...𝑁))
194		1red 10093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → 1 ∈ ℝ)
195		ltle 10164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℝ ∧ (2^nd ‘𝑇) ∈ ℝ) → (𝑦 < (2^nd ‘𝑇) → 𝑦 ≤ (2^nd ‘𝑇)))
196	116, 18, 195	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^nd ‘𝑇) → 𝑦 ≤ (2^nd ‘𝑇)))
197	196	imp 444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → 𝑦 ≤ (2^nd ‘𝑇))
198	124, 125, 194, 197	leadd1dd 10679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (𝑦 + 1) ≤ ((2^nd ‘𝑇) + 1))
199	60	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘𝑇) + 1) ≤ 𝑁)
200	57	peano2zd 11523	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℤ)
201	200	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) + 1) ∈ ℤ)
202		elfz 12370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((2^nd ‘𝑇) + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2^nd ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≤ 𝑁)))
203	201, 188, 189, 202	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≤ 𝑁)))
204	203	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((2^nd ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≤ 𝑁)))
205	198, 199, 204	mpbir2and 977	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁))
206		prssi 4385	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
207	193, 205, 206	syl2anc 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
208		imass2 5536	. . . . . . . . . . . . . . . . . 18 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
209	207, 208	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
210	177, 209	eqsstr3d 3673	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
211	174, 210	eqssd 3653	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
212		f1ofo 6182	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
213		forn 6156	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
214	26, 212, 213	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
215	214, 27	syl6eq 2701	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
216	215	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
217	211, 216	eqtrd 2685	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
218		undif 4082	. . . . . . . . . . . . . . . . 17 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) ↔ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
219	207, 218	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
220	219	imaeq2d 5501	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)))
221		fnresi 6046	. . . . . . . . . . . . . . . . . . 19 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
222		incom 3838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
223	222, 32	eqtri 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅
224		fnimadisj 6050	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ (((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅)
225	221, 223, 224	mp2an 708	. . . . . . . . . . . . . . . . . 18 ⊢ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅
226	225	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅)
227		nnuz 11761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
228	186, 227	syl6eleq 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
229		fzss1 12418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
230	228, 229	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
231	230	ssdifd 3779	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
232		resiima 5515	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
233	231, 232	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
234	233	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
235	226, 234	uneq12d 3801	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
236		imaundi 5580	. . . . . . . . . . . . . . . 16 ⊢ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
237		uncom 3790	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ∅)
238		un0 4000	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ∅) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
239	237, 238	eqtr2i 2674	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
240	235, 236, 239	3eqtr4g 2710	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
241	220, 240	eqtr3d 2687	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)) = (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
242	217, 241	uneq12d 3801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((𝑦 + 1)...𝑁))) = ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
243	172, 242	syl5eq 2697	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
244	243, 219	eqtrd 2685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = ((𝑦 + 1)...𝑁))
245	244	imaeq2d 5501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
246	171, 245	syl5eq 2697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
247	246	xpeq1d 5172	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
248	170, 247	uneq12d 3801	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
249	248	oveq2d 6706	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
250		iftrue 4125	. . . . . . . . 9 ⊢ (𝑦 < (2^nd ‘𝑇) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
251	250	csbeq1d 3573	. . . . . . . 8 ⊢ (𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
252		vex 3234	. . . . . . . . 9 ⊢ 𝑦 ∈ V
253		oveq2 6698	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
254	253	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)))
255	254	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}))
256		oveq1 6697	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
257	256	oveq1d 6705	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
258	257	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)))
259	258	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))
260	255, 259	uneq12d 3801	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})))
261	260	oveq2d 6706	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
262	252, 261	csbie 3592	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})))
263	251, 262	syl6eq 2701	. . . . . . 7 ⊢ (𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
264	263	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
265	250	csbeq1d 3573	. . . . . . . 8 ⊢ (𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
266	253	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)))
267	266	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}))
268	257	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
269	268	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
270	267, 269	uneq12d 3801	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
271	270	oveq2d 6706	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
272	252, 271	csbie 3592	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
273	265, 272	syl6eq 2701	. . . . . . 7 ⊢ (𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
274	273	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
275	249, 264, 274	3eqtr4d 2695	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
276		lenlt 10154	. . . . . . . . . 10 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
277	18, 116, 276	syl2an 493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
278	277	biimpar 501	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → (2^nd ‘𝑇) ≤ 𝑦)
279		imaco 5678	. . . . . . . . . . 11 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) = ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...(𝑦 + 1))))
280		imaundir 5581	. . . . . . . . . . . . . 14 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...(𝑦 + 1))) = (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...(𝑦 + 1))))
281		imassrn 5512	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}
282	281	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
283	176	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
284	17	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) ∈ ℕ)
285	18	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) ∈ ℝ)
286	116	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ)
287	186	nnred 11073	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
288	287	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ)
289		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) ≤ 𝑦)
290	116	lep1d 10993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1))
291	290	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1))
292	285, 286, 288, 289, 291	letrd 10232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) ≤ (𝑦 + 1))
293		fznn 12446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 + 1) ∈ ℤ → ((2^nd ‘𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2^nd ‘𝑇) ∈ ℕ ∧ (2^nd ‘𝑇) ≤ (𝑦 + 1))))
294	187, 293	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2^nd ‘𝑇) ∈ ℕ ∧ (2^nd ‘𝑇) ≤ (𝑦 + 1))))
295	294	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2^nd ‘𝑇) ∈ ℕ ∧ (2^nd ‘𝑇) ≤ (𝑦 + 1))))
296	284, 292, 295	mpbir2and 977	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) ∈ (1...(𝑦 + 1)))
297	50	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) + 1) ∈ ℕ)
298		1red 10093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → 1 ∈ ℝ)
299	285, 286, 298, 289	leadd1dd 10679	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) + 1) ≤ (𝑦 + 1))
300		fznn 12446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 + 1) ∈ ℤ → (((2^nd ‘𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2^nd ‘𝑇) + 1) ∈ ℕ ∧ ((2^nd ‘𝑇) + 1) ≤ (𝑦 + 1))))
301	187, 300	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2^nd ‘𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2^nd ‘𝑇) + 1) ∈ ℕ ∧ ((2^nd ‘𝑇) + 1) ≤ (𝑦 + 1))))
302	301	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((2^nd ‘𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2^nd ‘𝑇) + 1) ∈ ℕ ∧ ((2^nd ‘𝑇) + 1) ≤ (𝑦 + 1))))
303	297, 299, 302	mpbir2and 977	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) + 1) ∈ (1...(𝑦 + 1)))
304		prssi 4385	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
305	296, 303, 304	syl2anc 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
306		imass2 5536	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))))
307	305, 306	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))))
308	283, 307	eqsstr3d 3673	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ⊆ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))))
309	282, 308	eqssd 3653	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) = ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
310	215	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
311	309, 310	eqtrd 2685	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
312		undif 4082	. . . . . . . . . . . . . . . . . 18 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) ↔ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...(𝑦 + 1)))
313	305, 312	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...(𝑦 + 1)))
314	313	imaeq2d 5501	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...(𝑦 + 1))))
315	225	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅)
316		eluzp1p1 11751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
317	150, 316	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
318	317	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
319	149, 318	eqeltrrd 2731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
320		fzss2 12419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
321	319, 320	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
322	321	ssdifd 3779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
323	322	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
324		resiima 5515	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
325	323, 324	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
326	315, 325	uneq12d 3801	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
327		imaundi 5580	. . . . . . . . . . . . . . . . 17 ⊢ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
328		uncom 3790	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ∅)
329		un0 4000	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
330	328, 329	eqtr2i 2674	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
331	326, 327, 330	3eqtr4g 2710	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
332	314, 331	eqtr3d 2687	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...(𝑦 + 1))) = ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
333	311, 332	uneq12d 3801	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (1...(𝑦 + 1)))) = ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
334	280, 333	syl5eq 2697	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...(𝑦 + 1))) = ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
335	334, 313	eqtrd 2685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...(𝑦 + 1))) = (1...(𝑦 + 1)))
336	335	imaeq2d 5501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (1...(𝑦 + 1)))) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
337	279, 336	syl5eq 2697	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
338	337	xpeq1d 5172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
339		imaco 5678	. . . . . . . . . . 11 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)))
340	112	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
341		incom 3838	. . . . . . . . . . . . . . . 16 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
342	126	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) + 1) ∈ ℝ)
343	186	peano2nnd 11075	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
344	343	nnred 11073	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
345	344	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ)
346	19	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
347	116	ltp1d 10992	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
348	347	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1))
349	285, 286, 288, 289, 348	lelttrd 10233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) < (𝑦 + 1))
350	285, 288, 298, 349	ltadd1dd 10676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) + 1) < ((𝑦 + 1) + 1))
351	285, 342, 345, 346, 350	lttrd 10236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (2^nd ‘𝑇) < ((𝑦 + 1) + 1))
352		ltnle 10155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → ((2^nd ‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^nd ‘𝑇)))
353	18, 344, 352	syl2an 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^nd ‘𝑇)))
354	353	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^nd ‘𝑇)))
355	351, 354	mpbid 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2^nd ‘𝑇))
356		elfzle1 12382	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2^nd ‘𝑇))
357	355, 356	nsyl 135	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ¬ (2^nd ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
358		disjsn 4278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇)}) = ∅ ↔ ¬ (2^nd ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
359	357, 358	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇)}) = ∅)
360		ltnle 10155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → (((2^nd ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
361	126, 344, 360	syl2an 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
362	361	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((2^nd ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
363	350, 362	mpbid 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
364		elfzle1 12382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
365	363, 364	nsyl 135	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ¬ ((2^nd ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
366		disjsn 4278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑦 + 1) + 1)...𝑁) ∩ {((2^nd ‘𝑇) + 1)}) = ∅ ↔ ¬ ((2^nd ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
367	365, 366	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^nd ‘𝑇) + 1)}) = ∅)
368	359, 367	uneq12d 3801	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^nd ‘𝑇) + 1)})) = (∅ ∪ ∅))
369	140	ineq2i 3844	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)}))
370		indi 3906	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})) = (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^nd ‘𝑇) + 1)}))
371	369, 370	eqtr2i 2674	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^nd ‘𝑇) + 1)})) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
372	368, 371, 144	3eqtr3g 2708	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅)
373	341, 372	syl5eq 2697	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
374		fnimadisj 6050	. . . . . . . . . . . . . . 15 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅)
375	340, 373, 374	syl2anc 694	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅)
376	343, 227	syl6eleq 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
377		fzss1 12418	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
378		reldisj 4053	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
379	376, 377, 378	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
380	379	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
381	372, 380	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
382		resiima 5515	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
383	381, 382	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
384	375, 383	uneq12d 3801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁))) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁)))
385		imaundir 5581	. . . . . . . . . . . . 13 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)) = (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)))
386		uncom 3790	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅)
387		un0 4000	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁)
388	386, 387	eqtr2i 2674	. . . . . . . . . . . . 13 ⊢ (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁))
389	384, 385, 388	3eqtr4g 2710	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
390	389	imaeq2d 5501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((2^nd ‘(1^st ‘𝑇)) “ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
391	339, 390	syl5eq 2697	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
392	391	xpeq1d 5172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
393	338, 392	uneq12d 3801	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^nd ‘𝑇) ≤ 𝑦) → (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
394	278, 393	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
395	394	oveq2d 6706	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
396		iffalse 4128	. . . . . . . . 9 ⊢ (¬ 𝑦 < (2^nd ‘𝑇) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
397	396	csbeq1d 3573	. . . . . . . 8 ⊢ (¬ 𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
398		ovex 6718	. . . . . . . . 9 ⊢ (𝑦 + 1) ∈ V
399		oveq2 6698	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
400	399	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))))
401	400	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}))
402		oveq1 6697	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
403	402	oveq1d 6705	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
404	403	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)))
405	404	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
406	401, 405	uneq12d 3801	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
407	406	oveq2d 6706	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
408	398, 407	csbie 3592	. . . . . . . 8 ⊢ ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
409	397, 408	syl6eq 2701	. . . . . . 7 ⊢ (¬ 𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
410	409	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
411	396	csbeq1d 3573	. . . . . . . 8 ⊢ (¬ 𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
412	399	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
413	412	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
414	403	imaeq2d 5501	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
415	414	xpeq1d 5172	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
416	413, 415	uneq12d 3801	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
417	416	oveq2d 6706	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
418	398, 417	csbie 3592	. . . . . . . 8 ⊢ ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
419	411, 418	syl6eq 2701	. . . . . . 7 ⊢ (¬ 𝑦 < (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
420	419	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
421	395, 410, 420	3eqtr4d 2695	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
422	275, 421	pm2.61dan 849	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
423	422	mpteq2dva 4777	. . 3 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
424	109, 423	eqtr4d 2688	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))))
425		opex 4962	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∈ V
426	425, 22	op1std 7220	. . . . . 6 ⊢ (𝑡 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ → (1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩)
427	425, 22	op2ndd 7221	. . . . . 6 ⊢ (𝑡 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ → (2^nd ‘𝑡) = (2^nd ‘𝑇))
428		breq2 4689	. . . . . . . . 9 ⊢ ((2^nd ‘𝑡) = (2^nd ‘𝑇) → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
429	428	ifbid 4141	. . . . . . . 8 ⊢ ((2^nd ‘𝑡) = (2^nd ‘𝑇) → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
430	429	csbeq1d 3573	. . . . . . 7 ⊢ ((2^nd ‘𝑡) = (2^nd ‘𝑇) → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
431		fvex 6239	. . . . . . . . . 10 ⊢ (1^st ‘(1^st ‘𝑇)) ∈ V
432	431, 79	op1std 7220	. . . . . . . . 9 ⊢ ((1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
433	431, 79	op2ndd 7221	. . . . . . . . 9 ⊢ ((1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ → (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
434		id 22	. . . . . . . . . 10 ⊢ ((1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)) → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
435		imaeq1 5496	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)))
436	435	xpeq1d 5172	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}))
437		imaeq1 5496	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)))
438	437	xpeq1d 5172	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))
439	436, 438	uneq12d 3801	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))
440	434, 439	oveqan12d 6709	. . . . . . . . 9 ⊢ (((1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) → ((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
441	432, 433, 440	syl2anc 694	. . . . . . . 8 ⊢ ((1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ → ((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
442	441	csbeq2dv 4025	. . . . . . 7 ⊢ ((1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
443	430, 442	sylan9eqr 2707	. . . . . 6 ⊢ (((1^st ‘𝑡) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∧ (2^nd ‘𝑡) = (2^nd ‘𝑇)) → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
444	426, 427, 443	syl2anc 694	. . . . 5 ⊢ (𝑡 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
445	444	mpteq2dv 4778	. . . 4 ⊢ (𝑡 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))))
446	445	eqeq2d 2661	. . 3 ⊢ (𝑡 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
447	446, 3	elrab2 3399	. 2 ⊢ (⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ 𝑆 ↔ (⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + (((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
448	89, 424, 447	sylanbrc 699	1 ⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {cab 2637 ≠ wne 2823 {crab 2945 Vcvv 3231 ⦋csb 3566 ∖ cdif 3604 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 ifcif 4119 {csn 4210 {cpr 4212 ⟨cop 4216 class class class wbr 4685 ↦ cmpt 4762 I cid 5052 × cxp 5141 ran crn 5144 ↾ cres 5145 “ cima 5146 ∘ ccom 5147 Fn wfn 5921 ⟶wf 5922 –onto→wfo 5924 –1-1-onto→wf1o 5925 ‘cfv 5926 (class class class)co 6690 ∘_𝑓 cof 6937 1^st c1st 7208 2^nd c2nd 7209 ↑_𝑚 cmap 7899 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 < clt 10112 ≤ cle 10113 − cmin 10304 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ℤ_≥cuz 11725 ...cfz 12364 ..^cfzo 12504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365

This theorem is referenced by: poimirlem22 33561

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