Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem8 Structured version   Visualization version   GIF version

Theorem poimirlem8 33730
Description: Lemma for poimir 33755, establishing that away from the opposite vertex the walks in poimirlem9 33731 yield the same vertices. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem9.1 (𝜑𝑇𝑆)
poimirlem9.2 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
poimirlem9.3 (𝜑𝑈𝑆)
Assertion
Ref Expression
poimirlem8 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝑈,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑈,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑡,𝑈   𝑆,𝑗,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem8
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimirlem9.3 . . . . . . . 8 (𝜑𝑈𝑆)
2 elrabi 3499 . . . . . . . . 9 (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3 poimirlem22.s . . . . . . . . 9 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
42, 3eleq2s 2857 . . . . . . . 8 (𝑈𝑆𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
51, 4syl 17 . . . . . . 7 (𝜑𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6 xp1st 7365 . . . . . . 7 (𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
75, 6syl 17 . . . . . 6 (𝜑 → (1st𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8 xp2nd 7366 . . . . . 6 ((1st𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
97, 8syl 17 . . . . 5 (𝜑 → (2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10 fvex 6362 . . . . . 6 (2nd ‘(1st𝑈)) ∈ V
11 f1oeq1 6288 . . . . . 6 (𝑓 = (2nd ‘(1st𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3490 . . . . 5 ((2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 208 . . . 4 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1ofn 6299 . . . 4 ((2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑈)) Fn (1...𝑁))
1513, 14syl 17 . . 3 (𝜑 → (2nd ‘(1st𝑈)) Fn (1...𝑁))
16 difss 3880 . . 3 ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)
17 fnssres 6165 . . 3 (((2nd ‘(1st𝑈)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)) → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
1815, 16, 17sylancl 697 . 2 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
19 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
20 elrabi 3499 . . . . . . . . 9 (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2120, 3eleq2s 2857 . . . . . . . 8 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2219, 21syl 17 . . . . . . 7 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23 xp1st 7365 . . . . . . 7 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2422, 23syl 17 . . . . . 6 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25 xp2nd 7366 . . . . . 6 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
2624, 25syl 17 . . . . 5 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
27 fvex 6362 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
28 f1oeq1 6288 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2927, 28elab 3490 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
3026, 29sylib 208 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31 f1ofn 6299 . . . 4 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
3230, 31syl 17 . . 3 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
33 fnssres 6165 . . 3 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)) → ((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
3432, 16, 33sylancl 697 . 2 (𝜑 → ((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
35 poimirlem9.2 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
36 fzp1elp1 12587 . . . . . . . . . . . 12 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
3735, 36syl 17 . . . . . . . . . . 11 (𝜑 → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
38 poimir.0 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℕ)
3938nncnd 11228 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℂ)
40 npcan1 10647 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
4139, 40syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4241oveq2d 6829 . . . . . . . . . . 11 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
4337, 42eleqtrd 2841 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
44 fzsplit 12560 . . . . . . . . . 10 (((2nd𝑇) + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
4543, 44syl 17 . . . . . . . . 9 (𝜑 → (1...𝑁) = ((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
4645difeq1d 3870 . . . . . . . 8 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
47 difundir 4023 . . . . . . . . 9 (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
48 elfznn 12563 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
4935, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℕ)
5049nncnd 11228 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑇) ∈ ℂ)
51 npcan1 10647 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
53 nnuz 11916 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
5449, 53syl6eleq 2849 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘1))
5552, 54eqeltrd 2839 . . . . . . . . . . . . . 14 (𝜑 → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘1))
5649nnzd 11673 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd𝑇) ∈ ℤ)
57 peano2zm 11612 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
5856, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
59 uzid 11894 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) − 1) ∈ ℤ → ((2nd𝑇) − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
60 peano2uz 11934 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) − 1) ∈ (ℤ‘((2nd𝑇) − 1)) → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6158, 59, 603syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6252, 61eqeltrrd 2840 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)))
63 peano2uz 11934 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6462, 63syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
65 fzsplit2 12559 . . . . . . . . . . . . . 14 (((((2nd𝑇) − 1) + 1) ∈ (ℤ‘1) ∧ ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1))) → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))))
6655, 64, 65syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))))
6752oveq1d 6828 . . . . . . . . . . . . . . 15 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = ((2nd𝑇)...((2nd𝑇) + 1)))
68 fzpr 12589 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℤ → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
6956, 68syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7067, 69eqtrd 2794 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7170uneq2d 3910 . . . . . . . . . . . . 13 (𝜑 → ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7266, 71eqtrd 2794 . . . . . . . . . . . 12 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7372difeq1d 3870 . . . . . . . . . . 11 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
7449nnred 11227 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
7574ltm1d 11148 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
7658zred 11674 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
7776, 74ltnled 10376 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1)))
7875, 77mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1))
79 elfzle2 12538 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (1...((2nd𝑇) − 1)) → (2nd𝑇) ≤ ((2nd𝑇) − 1))
8078, 79nsyl 135 . . . . . . . . . . . . . 14 (𝜑 → ¬ (2nd𝑇) ∈ (1...((2nd𝑇) − 1)))
81 difsn 4473 . . . . . . . . . . . . . 14 (¬ (2nd𝑇) ∈ (1...((2nd𝑇) − 1)) → ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) = (1...((2nd𝑇) − 1)))
8280, 81syl 17 . . . . . . . . . . . . 13 (𝜑 → ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) = (1...((2nd𝑇) − 1)))
83 peano2re 10401 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd𝑇) + 1) ∈ ℝ)
8474, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
8574ltp1d 11146 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
8676, 74, 84, 75, 85lttrd 10390 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < ((2nd𝑇) + 1))
8776, 84ltnled 10376 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1)))
8886, 87mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
89 elfzle2 12538 . . . . . . . . . . . . . . 15 (((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
9088, 89nsyl 135 . . . . . . . . . . . . . 14 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)))
91 difsn 4473 . . . . . . . . . . . . . 14 (¬ ((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)) → ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
9290, 91syl 17 . . . . . . . . . . . . 13 (𝜑 → ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
9382, 92ineq12d 3958 . . . . . . . . . . . 12 (𝜑 → (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))))
94 difun2 4192 . . . . . . . . . . . . 13 (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
95 df-pr 4324 . . . . . . . . . . . . . 14 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
9695difeq2i 3868 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
97 difundi 4022 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)}))
9894, 96, 973eqtrri 2787 . . . . . . . . . . . 12 (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
99 inidm 3965 . . . . . . . . . . . 12 ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))) = (1...((2nd𝑇) − 1))
10093, 98, 993eqtr3g 2817 . . . . . . . . . . 11 (𝜑 → (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
10173, 100eqtrd 2794 . . . . . . . . . 10 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
102 peano2re 10401 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) + 1) ∈ ℝ → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10384, 102syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10484ltp1d 11146 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
10574, 84, 103, 85, 104lttrd 10390 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) < (((2nd𝑇) + 1) + 1))
10674, 103ltnled 10376 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇)))
107105, 106mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
108 elfzle1 12537 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
109107, 108nsyl 135 . . . . . . . . . . . . 13 (𝜑 → ¬ (2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
110 difsn 4473 . . . . . . . . . . . . 13 (¬ (2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
111109, 110syl 17 . . . . . . . . . . . 12 (𝜑 → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
11284, 103ltnled 10376 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1)))
113104, 112mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
114 elfzle1 12537 . . . . . . . . . . . . . 14 (((2nd𝑇) + 1) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
115113, 114nsyl 135 . . . . . . . . . . . . 13 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
116 difsn 4473 . . . . . . . . . . . . 13 (¬ ((2nd𝑇) + 1) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
117115, 116syl 17 . . . . . . . . . . . 12 (𝜑 → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
118111, 117ineq12d 3958 . . . . . . . . . . 11 (𝜑 → ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)))
11995difeq2i 3868 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
120 difundi 4022 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)}))
121119, 120eqtr2i 2783 . . . . . . . . . . 11 ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
122 inidm 3965 . . . . . . . . . . 11 (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)) = ((((2nd𝑇) + 1) + 1)...𝑁)
123118, 121, 1223eqtr3g 2817 . . . . . . . . . 10 (𝜑 → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
124101, 123uneq12d 3911 . . . . . . . . 9 (𝜑 → (((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12547, 124syl5eq 2806 . . . . . . . 8 (𝜑 → (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12646, 125eqtrd 2794 . . . . . . 7 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
127126eleq2d 2825 . . . . . 6 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁))))
128 elun 3896 . . . . . 6 (𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
129127, 128syl6bb 276 . . . . 5 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))))
130129biimpa 502 . . . 4 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
131 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
132131breq2d 4816 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
133132ifbid 4252 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
134133csbeq1d 3681 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
135 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
136135fveq2d 6356 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
137135fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
138137imaeq1d 5623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
139138xpeq1d 5295 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
140137imaeq1d 5623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
141140xpeq1d 5295 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
142139, 141uneq12d 3911 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
143136, 142oveq12d 6831 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
144143csbeq2dv 4135 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
145134, 144eqtrd 2794 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
146145mpteq2dv 4897 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
147146eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
148147, 3elrab2 3507 . . . . . . . . . . . . . . 15 (𝑇𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
149148simprbi 483 . . . . . . . . . . . . . 14 (𝑇𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
15019, 149syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
151 xp1st 7365 . . . . . . . . . . . . . . . 16 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
15224, 151syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
153 elmapi 8045 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
154152, 153syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
155 elfzoelz 12664 . . . . . . . . . . . . . . 15 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
156155ssriv 3748 . . . . . . . . . . . . . 14 (0..^𝐾) ⊆ ℤ
157 fss 6217 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
158154, 156, 157sylancl 697 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
15938, 150, 158, 30, 35poimirlem1 33723 . . . . . . . . . . . 12 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
16038adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
161 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (2nd𝑡) = (2nd𝑈))
162161breq2d 4816 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑈)))
163162ifbid 4252 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑈 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)))
164163csbeq1d 3681 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑈if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
165 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (1st𝑡) = (1st𝑈))
166165fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑈)))
167165fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑈 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑈)))
168167imaeq1d 5623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑗)))
169168xpeq1d 5295 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}))
170167imaeq1d 5623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)))
171170xpeq1d 5295 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
172169, 171uneq12d 3911 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))
173166, 172oveq12d 6831 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑈 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
174173csbeq2dv 4135 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑈if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
175164, 174eqtrd 2794 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑈if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
176175mpteq2dv 4897 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
177176eqeq2d 2770 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
178177, 3elrab2 3507 . . . . . . . . . . . . . . . . . 18 (𝑈𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
179178simprbi 483 . . . . . . . . . . . . . . . . 17 (𝑈𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
1801, 179syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
181180adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘𝑓 + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
182 xp1st 7365 . . . . . . . . . . . . . . . . . . 19 ((1st𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
1837, 182syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
184 elmapi 8045 . . . . . . . . . . . . . . . . . 18 ((1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
185183, 184syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
186 fss 6217 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
187185, 156, 186sylancl 697 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
188187adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
18913adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
19035adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
191 xp2nd 7366 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑈) ∈ (0...𝑁))
1925, 191syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑈) ∈ (0...𝑁))
193 eldifsn 4462 . . . . . . . . . . . . . . . . 17 ((2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)))
194193biimpri 218 . . . . . . . . . . . . . . . 16 (((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
195192, 194sylan 489 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
196160, 181, 188, 189, 190, 195poimirlem2 33724 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
197196ex 449 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑈) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
198197necon1bd 2950 . . . . . . . . . . . 12 (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑈) = (2nd𝑇)))
199159, 198mpd 15 . . . . . . . . . . 11 (𝜑 → (2nd𝑈) = (2nd𝑇))
200199oveq1d 6828 . . . . . . . . . 10 (𝜑 → ((2nd𝑈) − 1) = ((2nd𝑇) − 1))
201200oveq2d 6829 . . . . . . . . 9 (𝜑 → (1...((2nd𝑈) − 1)) = (1...((2nd𝑇) − 1)))
202201eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑘 ∈ (1...((2nd𝑈) − 1)) ↔ 𝑘 ∈ (1...((2nd𝑇) − 1))))
203202biimpar 503 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
20438adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑁 ∈ ℕ)
2051adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑈𝑆)
206199, 35eqeltrd 2839 . . . . . . . . 9 (𝜑 → (2nd𝑈) ∈ (1...(𝑁 − 1)))
207206adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
208 simpr 479 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
209204, 3, 205, 207, 208poimirlem6 33728 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
210203, 209syldan 488 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
21138adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑁 ∈ ℕ)
21219adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑇𝑆)
21335adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
214 simpr 479 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑇) − 1)))
215211, 3, 212, 213, 214poimirlem6 33728 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
216210, 215eqtr3d 2796 . . . . 5 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
217199oveq1d 6828 . . . . . . . . . . 11 (𝜑 → ((2nd𝑈) + 1) = ((2nd𝑇) + 1))
218217oveq1d 6828 . . . . . . . . . 10 (𝜑 → (((2nd𝑈) + 1) + 1) = (((2nd𝑇) + 1) + 1))
219218oveq1d 6828 . . . . . . . . 9 (𝜑 → ((((2nd𝑈) + 1) + 1)...𝑁) = ((((2nd𝑇) + 1) + 1)...𝑁))
220219eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
221220biimpar 503 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
22238adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
2231adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑈𝑆)
224206adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
225 simpr 479 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
226222, 3, 223, 224, 225poimirlem7 33729 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
227221, 226syldan 488 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
22838adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
22919adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑇𝑆)
23035adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
231 simpr 479 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
232228, 3, 229, 230, 231poimirlem7 33729 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
233227, 232eqtr3d 2796 . . . . 5 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
234216, 233jaodan 861 . . . 4 ((𝜑 ∧ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
235130, 234syldan 488 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
236 fvres 6368 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
237236adantl 473 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
238 fvres 6368 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
239238adantl 473 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
240235, 237, 2393eqtr4d 2804 . 2 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘))
24118, 34, 240eqfnfvd 6477 1 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1632  wcel 2139  {cab 2746  wne 2932  ∃*wrmo 3053  {crab 3054  csb 3674  cdif 3712  cun 3713  cin 3714  wss 3715  ifcif 4230  {csn 4321  {cpr 4323   class class class wbr 4804  cmpt 4881   × cxp 5264  cres 5268  cima 5269   Fn wfn 6044  wf 6045  1-1-ontowf1o 6048  cfv 6049  crio 6773  (class class class)co 6813  𝑓 cof 7060  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   < clt 10266  cle 10267  cmin 10458  cn 11212  2c2 11262  cz 11569  cuz 11879  ...cfz 12519  ..^cfzo 12659
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 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  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-rmo 3058  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 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660
This theorem is referenced by:  poimirlem9  33731
  Copyright terms: Public domain W3C validator