poimirlem23 - Mathbox for Brendan Leahy

	Mathbox for Brendan Leahy		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > poimirlem23		Structured version Visualization version GIF version

Theorem poimirlem23 33763

Description: Lemma for poimir 33773, two ways of expressing the property that a face is not on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem23.1	⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
poimirlem23.2	⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem23.3	⊢ (𝜑 → 𝑉 ∈ (0...𝑁))

**Assertion**
Ref	Expression
poimirlem23	⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))

Distinct variable groups: 𝑗,𝑝,𝑦,𝜑 𝑗,𝑁,𝑦 𝑇,𝑗,𝑦 𝑈,𝑗,𝑦 𝑗,𝑉,𝑦 𝜑,𝑝 𝑗,𝐾,𝑝 𝑁,𝑝 𝑇,𝑝 𝑈,𝑝 𝑦,𝐾 𝑉,𝑝

**Proof of Theorem poimirlem23**
Step	Hyp	Ref	Expression
1		ovex 6842	. . . . . 6 ⊢ (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
2	1	csbex 4945	. . . . 5 ⊢ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
3	2	rgenw 3062	. . . 4 ⊢ ∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
4		eqid 2760	. . . . 5 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
5		fveq1 6352	. . . . . . 7 ⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝‘𝑁) = (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
6	5	neeq1d 2991	. . . . . 6 ⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑁) ≠ 0 ↔ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
7		df-ne 2933	. . . . . 6 ⊢ ((⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
8	6, 7	syl6bb 276	. . . . 5 ⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑁) ≠ 0 ↔ ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
9	4, 8	rexrnmpt 6533	. . . 4 ⊢ (∀𝑦 ∈ (0...(𝑁 − 1))⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
10	3, 9	ax-mp 5	. . 3 ⊢ (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
11		rexnal 3133	. . 3 ⊢ (∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
12	10, 11	bitri 264	. 2 ⊢ (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
13		poimir.0	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
14	13	nnzd 11693	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15		poimirlem23.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (0...𝑁))
16		elfzelz 12555	. . . . . . . . . . 11 ⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ ℤ)
18		zlem1lt 11641	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑁 ≤ 𝑉 ↔ (𝑁 − 1) < 𝑉))
19	14, 17, 18	syl2anc 696	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ≤ 𝑉 ↔ (𝑁 − 1) < 𝑉))
20		elfzle2 12558	. . . . . . . . . . 11 ⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ≤ 𝑁)
21	15, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ≤ 𝑁)
22	17	zred 11694	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ ℝ)
23	13	nnred 11247	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
24	22, 23	letri3d 10391	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 = 𝑁 ↔ (𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉)))
25	24	biimprd 238	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉) → 𝑉 = 𝑁))
26	21, 25	mpand 713	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ≤ 𝑉 → 𝑉 = 𝑁))
27	19, 26	sylbird 250	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) < 𝑉 → 𝑉 = 𝑁))
28	27	necon3ad 2945	. . . . . . 7 ⊢ (𝜑 → (𝑉 ≠ 𝑁 → ¬ (𝑁 − 1) < 𝑉))
29		nnm1nn0 11546	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
30	13, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
31		nn0fz0 12651	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℕ₀ ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
32	30, 31	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
33	32	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
34		iffalse 4239	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑁 − 1) < 𝑉 → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = ((𝑁 − 1) + 1))
35	13	nncnd 11248	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℂ)
36		npcan1 10667	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
37	35, 36	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
38	34, 37	sylan9eqr 2816	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = 𝑁)
39	38	csbeq1d 3681	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑁 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
40		oveq2 6822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
41	40	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑁 → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...𝑁)))
42	41	xpeq1d 5295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑁 → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑁)) × {1}))
43		oveq1 6821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
44	43	oveq1d 6829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
45	44	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑁 → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑁 + 1)...𝑁)))
46	45	xpeq1d 5295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑁 → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}))
47	42, 46	uneq12d 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑁 → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})))
48		poimirlem23.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
49		f1ofo 6306	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
50		foima 6282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
51	48, 49, 50	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
52	51	xpeq1d 5295	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑈 “ (1...𝑁)) × {1}) = ((1...𝑁) × {1}))
53	23	ltp1d 11166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
54	14	peano2zd 11697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
55		fzn 12570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
56	54, 14, 55	syl2anc 696	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
57	53, 56	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
58	57	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑈 “ ((𝑁 + 1)...𝑁)) = (𝑈 “ ∅))
59	58	xpeq1d 5295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ((𝑈 “ ∅) × {0}))
60		ima0 5639	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈 “ ∅) = ∅
61	60	xpeq1i 5292	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑈 “ ∅) × {0}) = (∅ × {0})
62		0xp 5356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ × {0}) = ∅
63	61, 62	eqtri 2782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑈 “ ∅) × {0}) = ∅
64	59, 63	syl6eq 2810	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ∅)
65	52, 64	uneq12d 3911	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
66		un0 4110	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
67	65, 66	syl6eq 2810	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
68	47, 67	sylan9eqr 2816	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
69	68	oveq2d 6830	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + ((1...𝑁) × {1})))
70	13, 69	csbied 3701	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ⦋𝑁 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + ((1...𝑁) × {1})))
71	70	adantr 472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋𝑁 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + ((1...𝑁) × {1})))
72	39, 71	eqtrd 2794	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + ((1...𝑁) × {1})))
73	72	fveq1d 6355	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇 ∘_𝑓 + ((1...𝑁) × {1}))‘𝑁))
74		elfzonn0 12727	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0..^𝐾) → 𝑗 ∈ ℕ₀)
75		nn0p1nn 11544	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
76	74, 75	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0..^𝐾) → (𝑗 + 1) ∈ ℕ)
77		elsni 4338	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {1} → 𝑦 = 1)
78	77	oveq2d 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {1} → (𝑗 + 𝑦) = (𝑗 + 1))
79	78	eleq1d 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {1} → ((𝑗 + 𝑦) ∈ ℕ ↔ (𝑗 + 1) ∈ ℕ))
80	76, 79	syl5ibrcom 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0..^𝐾) → (𝑦 ∈ {1} → (𝑗 + 𝑦) ∈ ℕ))
81	80	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1}) → (𝑗 + 𝑦) ∈ ℕ)
82	81	adantl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1})) → (𝑗 + 𝑦) ∈ ℕ)
83		poimirlem23.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))
84		1ex 10247	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
85	84	fconst 6252	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶{1}
86	85	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1...𝑁) × {1}):(1...𝑁)⟶{1})
87		ovexd 6844	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑁) ∈ V)
88		inidm 3965	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
89	82, 83, 86, 87, 87, 88	off 7078	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇 ∘_𝑓 + ((1...𝑁) × {1})):(1...𝑁)⟶ℕ)
90		elfz1end 12584	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
91	13, 90	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92	89, 91	ffvelrnd 6524	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑇 ∘_𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈ ℕ)
93	92	adantr 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ((𝑇 ∘_𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈ ℕ)
94	73, 93	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ∈ ℕ)
95	94	nnne0d 11277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0)
96		breq1 4807	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 < 𝑉 ↔ (𝑁 − 1) < 𝑉))
97		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑁 − 1) → 𝑦 = (𝑁 − 1))
98		oveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
99	96, 97, 98	ifbieq12d 4257	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑁 − 1) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)))
100	99	csbeq1d 3681	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑁 − 1) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
101	100	fveq1d 6355	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑁 − 1) → (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
102	101	neeq1d 2991	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑁 − 1) → ((⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
103	7, 102	syl5bbr 274	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑁 − 1) → (¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
104	103	rspcev 3449	. . . . . . . . . 10 ⊢ (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧ (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
105	33, 95, 104	syl2anc 696	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
106	105, 11	sylib 208	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
107	106	ex 449	. . . . . . 7 ⊢ (𝜑 → (¬ (𝑁 − 1) < 𝑉 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
108	28, 107	syld 47	. . . . . 6 ⊢ (𝜑 → (𝑉 ≠ 𝑁 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
109	108	necon4ad 2951	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 → 𝑉 = 𝑁))
110	109	pm4.71rd 670	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)))
111	30	nn0zd 11692	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
112		uzid 11914	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
113		peano2uz 11954	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
114	111, 112, 113	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
115	37, 114	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
116		fzss2 12594	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
117	115, 116	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
118	117	sselda 3744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ (0...𝑁))
119	91	adantr 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈ (1...𝑁))
120		ffn 6206	. . . . . . . . . . . . . . 15 ⊢ (𝑇:(1...𝑁)⟶(0..^𝐾) → 𝑇 Fn (1...𝑁))
121	83, 120	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 Fn (1...𝑁))
122	121	adantr 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 Fn (1...𝑁))
123	84	fconst 6252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1}
124		c0ex 10246	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
125	124	fconst 6252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}
126	123, 125	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0})
127		dff1o3 6305	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡𝑈))
128	127	simprbi 483	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑈)
129		imain 6135	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
130	48, 128, 129	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
131		elfzelz 12555	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
132	131	zred 11694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
133	132	ltp1d 11166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
134		fzdisj 12581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
135	133, 134	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
136	135	imaeq2d 5624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
137	136, 60	syl6eq 2810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
138	130, 137	sylan9req 2815	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
139		fun 6227	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
140	126, 138, 139	sylancr 698	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
141		elfznn0 12646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
142	141, 75	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
143		nnuz 11936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
144	142, 143	syl6eleq 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
145		elfzuz3 12552	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
146		fzsplit2 12579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
147	144, 145, 146	syl2anc 696	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
148	147	imaeq2d 5624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
149		imaundi 5703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))
150	148, 149	syl6req 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑁) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (𝑈 “ (1...𝑁)))
151	150, 51	sylan9eqr 2816	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
152	151	feq2d 6192	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
153	140, 152	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
154		ffn 6206	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁))
155	153, 154	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁))
156		ovexd 6844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ V)
157		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → (𝑇‘𝑁) = (𝑇‘𝑁))
158		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
159	122, 155, 156, 156, 88, 157, 158	ofval 7072	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)))
160	119, 159	mpdan 705	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)))
161	160	eqeq1d 2762	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0))
162	83, 91	ffvelrnd 6524	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇‘𝑁) ∈ (0..^𝐾))
163		elfzonn0 12727	. . . . . . . . . . . . . 14 ⊢ ((𝑇‘𝑁) ∈ (0..^𝐾) → (𝑇‘𝑁) ∈ ℕ₀)
164	162, 163	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑇‘𝑁) ∈ ℕ₀)
165	164	nn0red 11564	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇‘𝑁) ∈ ℝ)
166	165	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇‘𝑁) ∈ ℝ)
167	164	nn0ge0d 11566	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑇‘𝑁))
168	167	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ (𝑇‘𝑁))
169		1re 10251	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
170		snssi 4484	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ)
171	169, 170	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {1} ⊆ ℝ
172		0re 10252	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
173		snssi 4484	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
174	172, 173	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {0} ⊆ ℝ
175	171, 174	unssi 3931	. . . . . . . . . . . 12 ⊢ ({1} ∪ {0}) ⊆ ℝ
176	153, 119	ffvelrnd 6524	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}))
177	175, 176	sseldi 3742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ)
178		elun 3896	. . . . . . . . . . . . 13 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) ↔ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0}))
179		0le1 10763	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
180		elsni 4338	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 1)
181	179, 180	syl5breqr 4842	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
182		0le0 11322	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
183		elsni 4338	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
184	182, 183	syl5breqr 4842	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
185	181, 184	jaoi 393	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0}) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
186	178, 185	sylbi 207	. . . . . . . . . . . 12 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
187	176, 186	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
188		add20 10752	. . . . . . . . . . 11 ⊢ ((((𝑇‘𝑁) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑁)) ∧ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ ∧ 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))) → (((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
189	166, 168, 177, 187, 188	syl22anc 1478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
190	161, 189	bitrd 268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
191	118, 190	syldan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
192	191	ralbidva 3123	. . . . . . 7 ⊢ (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
193	192	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
194		breq2 4808	. . . . . . . . . . . . . 14 ⊢ (𝑉 = 𝑁 → (𝑦 < 𝑉 ↔ 𝑦 < 𝑁))
195	194	ifbid 4252	. . . . . . . . . . . . 13 ⊢ (𝑉 = 𝑁 → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if(𝑦 < 𝑁, 𝑦, (𝑦 + 1)))
196		elfzelz 12555	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
197	196	zred 11694	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
198	197	adantl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
199	30	nn0red 11564	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
200	199	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
201	23	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
202		elfzle2 12558	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
203	202	adantl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
204	23	ltm1d 11168	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
205	204	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
206	198, 200, 201, 203, 205	lelttrd 10407	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
207	206	iftrued 4238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑁, 𝑦, (𝑦 + 1)) = 𝑦)
208	195, 207	sylan9eqr 2816	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑉 = 𝑁) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦)
209	208	an32s 881	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦)
210	209	csbeq1d 3681	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
211	210	fveq1d 6355	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
212	211	eqeq1d 2762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
213	212	ralbidva 3123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
214		nfv 1992	. . . . . . . 8 ⊢ Ⅎ𝑦((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0
215		nfcsb1v 3690	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
216		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑁
217	215, 216	nffv 6360	. . . . . . . . 9 ⊢ Ⅎ𝑗(⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)
218	217	nfeq1 2916	. . . . . . . 8 ⊢ Ⅎ𝑗(⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0
219		csbeq1a 3683	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
220	219	fveq1d 6355	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
221	220	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
222	214, 218, 221	cbvral 3306	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋𝑦 / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
223	213, 222	syl6bbr 278	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
224		ne0i 4064	. . . . . . . . . 10 ⊢ ((𝑁 − 1) ∈ (0...(𝑁 − 1)) → (0...(𝑁 − 1)) ≠ ∅)
225		r19.3rzv 4208	. . . . . . . . . 10 ⊢ ((0...(𝑁 − 1)) ≠ ∅ → ((𝑇‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0))
226	32, 224, 225	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0))
227		elfzelz 12555	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℤ)
228	227	zred 11694	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℝ)
229	228	ltp1d 11166	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 < (𝑗 + 1))
230	229, 134	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
231	230	imaeq2d 5624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
232	231, 60	syl6eq 2810	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
233	130, 232	sylan9req 2815	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
234	233	adantlr 753	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
235		simplr 809	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) = 𝑁)
236		f1ofn 6300	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
237	48, 236	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 Fn (1...𝑁))
238	237	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑈 Fn (1...𝑁))
239		elfznn0 12646	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ₀)
240	239, 75	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ)
241	240, 143	syl6eleq 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ (ℤ_≥‘1))
242		fzss1 12593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘1) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
243	241, 242	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
244	243	adantl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
245	37	adantr 472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
246		elfzuz3 12552	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑗))
247		eluzp1p1 11925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑗) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
248	246, 247	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
249	248	adantl 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
250	245, 249	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑗 + 1)))
251		eluzfz2 12562	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑗 + 1)) → 𝑁 ∈ ((𝑗 + 1)...𝑁))
252	250, 251	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑗 + 1)...𝑁))
253		fnfvima 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 Fn (1...𝑁) ∧ ((𝑗 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑗 + 1)...𝑁)) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
254	238, 244, 252, 253	syl3anc 1477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
255	254	adantlr 753	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
256	235, 255	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
257		fnconstg 6254	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ V → ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)))
258	84, 257	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗))
259		fnconstg 6254	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)))
260	124, 259	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁))
261		fvun2 6433	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)) ∧ (((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
262	258, 260, 261	mp3an12 1563	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
263	234, 256, 262	syl2anc 696	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
264	124	fvconst2 6634	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0)
265	256, 264	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0)
266	263, 265	eqtrd 2794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
267	266	ralrimiva 3104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈‘𝑁) = 𝑁) → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
268	267	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
269	32	adantr 472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
270		ax-1ne0 10217	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 0
271		imain 6135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))))
272	48, 128, 271	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))))
273	204, 37	breqtrrd 4832	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑁 − 1) < ((𝑁 − 1) + 1))
274		fzdisj 12581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 − 1) < ((𝑁 − 1) + 1) → ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁)) = ∅)
275	273, 274	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁)) = ∅)
276	275	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = (𝑈 “ ∅))
277	276, 60	syl6eq 2810	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ∅)
278	272, 277	eqtr3d 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅)
279	278	adantr 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅)
280	91	adantr 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ (1...𝑁))
281		elimasni 5650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ (𝑈 “ {𝑁}) → 𝑁𝑈𝑁)
282		fnbrfvb 6398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑈 Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑈‘𝑁) = 𝑁 ↔ 𝑁𝑈𝑁))
283	237, 91, 282	syl2anc 696	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 ↔ 𝑁𝑈𝑁))
284	281, 283	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑁 ∈ (𝑈 “ {𝑁}) → (𝑈‘𝑁) = 𝑁))
285	284	necon3ad 2945	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ¬ 𝑁 ∈ (𝑈 “ {𝑁})))
286	285	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ¬ 𝑁 ∈ (𝑈 “ {𝑁}))
287	280, 286	eldifd 3726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ ((1...𝑁) ∖ (𝑈 “ {𝑁})))
288		imadif 6134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})))
289	48, 128, 288	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})))
290		difun2 4192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
291	13, 143	syl6eleq 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
292		fzm1 12633	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)))
293	291, 292	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)))
294		elun 3896	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁}))
295		velsn 4337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ {𝑁} ↔ 𝑗 = 𝑁)
296	295	orbi2i 542	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))
297	294, 296	bitri 264	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))
298	293, 297	syl6rbbr 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ 𝑗 ∈ (1...𝑁)))
299	298	eqrdv 2758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
300	299	difeq1d 3870	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...𝑁) ∖ {𝑁}))
301	199, 23	ltnled 10396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
302	204, 301	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
303		elfzle2 12558	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
304	302, 303	nsyl 135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
305		difsn 4473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
306	304, 305	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
307	290, 300, 306	3eqtr3a 2818	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
308	307	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
309	51	difeq1d 3870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})) = ((1...𝑁) ∖ (𝑈 “ {𝑁})))
310	289, 308, 309	3eqtr3rd 2803	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
311	310	adantr 472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
312	287, 311	eleqtrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))))
313		fnconstg 6254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ V → ((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))))
314	84, 313	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1)))
315		fnconstg 6254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ V → ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))
316	124, 315	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁))
317		fvun1 6432	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))) ∧ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
318	314, 316, 317	mp3an12 1563	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1)))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
319	279, 312, 318	syl2anc 696	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
320	84	fvconst2 6634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1)
321	312, 320	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1)
322	319, 321	eqtrd 2794	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = 1)
323	322	neeq1d 2991	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ 1 ≠ 0))
324	270, 323	mpbiri 248	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0)
325		df-ne 2933	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
326		oveq2 6822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝑁 − 1) → (1...𝑗) = (1...(𝑁 − 1)))
327	326	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑁 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑁 − 1))))
328	327	xpeq1d 5295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑁 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑁 − 1))) × {1}))
329		oveq1 6821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = (𝑁 − 1) → (𝑗 + 1) = ((𝑁 − 1) + 1))
330	329	oveq1d 6829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝑁 − 1) → ((𝑗 + 1)...𝑁) = (((𝑁 − 1) + 1)...𝑁))
331	330	imaeq2d 5624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑁 − 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))
332	331	xpeq1d 5295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑁 − 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))
333	328, 332	uneq12d 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑁 − 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0})))
334	333	fveq1d 6355	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑁 − 1) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁))
335	334	neeq1d 2991	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑁 − 1) → (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0))
336	325, 335	syl5bbr 274	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑁 − 1) → (¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0))
337	336	rspcev 3449	. . . . . . . . . . . . . 14 ⊢ (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
338	269, 324, 337	syl2anc 696	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
339	338	ex 449	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
340		rexnal 3133	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
341	339, 340	syl6ib 241	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
342	341	necon4ad 2951	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 → (𝑈‘𝑁) = 𝑁))
343	268, 342	impbid 202	. . . . . . . . 9 ⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
344	226, 343	anbi12d 749	. . . . . . . 8 ⊢ (𝜑 → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
345		r19.26 3202	. . . . . . . 8 ⊢ (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
346	344, 345	syl6bbr 278	. . . . . . 7 ⊢ (𝜑 → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
347	346	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
348	193, 223, 347	3bitr4d 300	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))
349	348	pm5.32da 676	. . . 4 ⊢ (𝜑 → ((𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) ↔ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
350	110, 349	bitrd 268	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
351	350	notbid 307	. 2 ⊢ (𝜑 → (¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
352	12, 351	syl5bb 272	1 ⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 ∃wrex 3051 Vcvv 3340 ⦋csb 3674 ∖ cdif 3712 ∪ cun 3713 ∩ cin 3714 ⊆ wss 3715 ∅c0 4058 ifcif 4230 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 × cxp 5264 ^◡ccnv 5265 ran crn 5267 “ cima 5269 Fun wfun 6043 Fn wfn 6044 ⟶wf 6045 –onto→wfo 6047 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6814 ∘_𝑓 cof 7061 ℂcc 10146 ℝcr 10147 0cc0 10148 1c1 10149 + caddc 10151 < clt 10286 ≤ cle 10287 − cmin 10478 ℕcn 11232 ℕ₀cn0 11504 ℤcz 11589 ℤ_≥cuz 11899 ...cfz 12539 ..^cfzo 12679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-fzo 12680

This theorem is referenced by: poimirlem24 33764

W3C validator