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Theorem poimirlem8 35338
 Description: Lemma for poimir 35363, establishing that away from the opposite vertex the walks in poimirlem9 35339 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 3597 . . . . . . . . 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 2871 . . . . . . . 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 7726 . . . . . . 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 7727 . . . . . 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 6672 . . . . . 6 (2nd ‘(1st𝑈)) ∈ V
11 f1oeq1 6591 . . . . . 6 (𝑓 = (2nd ‘(1st𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3589 . . . . 5 ((2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 221 . . . 4 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1ofn 6604 . . . 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 4038 . . 3 ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)
17 fnssres 6454 . . 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 590 . 2 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
19 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
20 elrabi 3597 . . . . . . . . 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 2871 . . . . . . . 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 7726 . . . . . . 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 7727 . . . . . 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 6672 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
28 f1oeq1 6591 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2927, 28elab 3589 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
3026, 29sylib 221 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31 f1ofn 6604 . . . 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 6454 . . 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 590 . 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 13002 . . . . . . . . . . . 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 11683 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℂ)
40 npcan1 11096 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
4139, 40syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4241oveq2d 7167 . . . . . . . . . . 11 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
4337, 42eleqtrd 2855 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
44 fzsplit 12975 . . . . . . . . . 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 4028 . . . . . . . 8 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
47 difundir 4186 . . . . . . . . 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 12978 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
4935, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℕ)
5049nncnd 11683 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑇) ∈ ℂ)
51 npcan1 11096 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
53 nnuz 12314 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
5449, 53eleqtrdi 2863 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘1))
5552, 54eqeltrd 2853 . . . . . . . . . . . . . 14 (𝜑 → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘1))
5649nnzd 12118 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd𝑇) ∈ ℤ)
57 peano2zm 12057 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
5856, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
59 uzid 12290 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) − 1) ∈ ℤ → ((2nd𝑇) − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
60 peano2uz 12334 . . . . . . . . . . . . . . . . 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 2854 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)))
63 peano2uz 12334 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6462, 63syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
65 fzsplit2 12974 . . . . . . . . . . . . . 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 588 . . . . . . . . . . . . 13 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))))
6752oveq1d 7166 . . . . . . . . . . . . . . 15 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = ((2nd𝑇)...((2nd𝑇) + 1)))
68 fzpr 13004 . . . . . . . . . . . . . . . 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 4069 . . . . . . . . . . . . 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 4028 . . . . . . . . . . 11 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
7449nnred 11682 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
7574ltm1d 11603 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
7658zred 12119 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
7776, 74ltnled 10818 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1)))
7875, 77mpbid 235 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1))
79 elfzle2 12953 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (1...((2nd𝑇) − 1)) → (2nd𝑇) ≤ ((2nd𝑇) − 1))
8078, 79nsyl 142 . . . . . . . . . . . . . 14 (𝜑 → ¬ (2nd𝑇) ∈ (1...((2nd𝑇) − 1)))
81 difsn 4689 . . . . . . . . . . . . . 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 10844 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd𝑇) + 1) ∈ ℝ)
8474, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
8574ltp1d 11601 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
8676, 74, 84, 75, 85lttrd 10832 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < ((2nd𝑇) + 1))
8776, 84ltnled 10818 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1)))
8886, 87mpbid 235 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
89 elfzle2 12953 . . . . . . . . . . . . . . 15 (((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
9088, 89nsyl 142 . . . . . . . . . . . . . 14 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)))
91 difsn 4689 . . . . . . . . . . . . . 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 4119 . . . . . . . . . . . 12 (𝜑 → (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))))
94 difun2 4378 . . . . . . . . . . . . 13 (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
95 df-pr 4526 . . . . . . . . . . . . . 14 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
9695difeq2i 4026 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
97 difundi 4185 . . . . . . . . . . . . 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 4124 . . . . . . . . . . . 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 10844 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) + 1) ∈ ℝ → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10384, 102syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10484ltp1d 11601 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
10574, 84, 103, 85, 104lttrd 10832 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) < (((2nd𝑇) + 1) + 1))
10674, 103ltnled 10818 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇)))
107105, 106mpbid 235 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
108 elfzle1 12952 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
109107, 108nsyl 142 . . . . . . . . . . . . 13 (𝜑 → ¬ (2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
110 difsn 4689 . . . . . . . . . . . . 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 10818 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1)))
113104, 112mpbid 235 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
114 elfzle1 12952 . . . . . . . . . . . . . 14 (((2nd𝑇) + 1) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
115113, 114nsyl 142 . . . . . . . . . . . . 13 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
116 difsn 4689 . . . . . . . . . . . . 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 4119 . . . . . . . . . . 11 (𝜑 → ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)))
11995difeq2i 4026 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
120 difundi 4185 . . . . . . . . . . . 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 4124 . . . . . . . . . . 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 4070 . . . . . . . . 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 2838 . . . . . 6 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁))))
128 elun 4055 . . . . . 6 (𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
129127, 128bitrdi 290 . . . . 5 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))))
130129biimpa 481 . . . 4 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
131 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
132131breq2d 5045 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
133132ifbid 4444 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
134133csbeq1d 3810 . . . . . . . . . . . . . . . . . . 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 2fveq3 6664 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
136 2fveq3 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
137136imaeq1d 5901 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
138137xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
139136imaeq1d 5901 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
140139xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
141138, 140uneq12d 4070 . . . . . . . . . . . . . . . . . . . . 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})))
142135, 141oveq12d 7169 . . . . . . . . . . . . . . . . . . . 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}))))
143142csbeq2dv 3813 . . . . . . . . . . . . . . . . . . 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}))))
144134, 143eqtrd 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}))))
145144mpteq2dv 5129 . . . . . . . . . . . . . . . . 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})))))
146145eqeq2d 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}))))))
147146, 3elrab2 3606 . . . . . . . . . . . . . . 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}))))))
148147simprbi 501 . . . . . . . . . . . . . 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})))))
14919, 148syl 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})))))
150 xp1st 7726 . . . . . . . . . . . . . . . 16 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
15124, 150syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
152 elmapi 8439 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
153151, 152syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
154 elfzoelz 13080 . . . . . . . . . . . . . . 15 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155154ssriv 3897 . . . . . . . . . . . . . 14 (0..^𝐾) ⊆ ℤ
156 fss 6513 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
157153, 155, 156sylancl 590 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
15838, 149, 157, 30, 35poimirlem1 35331 . . . . . . . . . . . 12 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
15938adantr 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
160 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (2nd𝑡) = (2nd𝑈))
161160breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑈)))
162161ifbid 4444 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑈 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)))
163162csbeq1d 3810 . . . . . . . . . . . . . . . . . . . . . 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}))))
164 2fveq3 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑈)))
165 2fveq3 6664 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑈 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑈)))
166165imaeq1d 5901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑗)))
167166xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}))
168165imaeq1d 5901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)))
169168xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
170167, 169uneq12d 4070 . . . . . . . . . . . . . . . . . . . . . . . 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})))
171164, 170oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . 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}))))
172171csbeq2dv 3813 . . . . . . . . . . . . . . . . . . . . . 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}))))
173163, 172eqtrd 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}))))
174173mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . 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})))))
175174eqeq2d 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}))))))
176175, 3elrab2 3606 . . . . . . . . . . . . . . . . . 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}))))))
177176simprbi 501 . . . . . . . . . . . . . . . . 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})))))
1781, 177syl 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})))))
179178adantr 485 . . . . . . . . . . . . . . 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})))))
180 xp1st 7726 . . . . . . . . . . . . . . . . . . 19 ((1st𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
1817, 180syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
182 elmapi 8439 . . . . . . . . . . . . . . . . . 18 ((1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
183181, 182syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
184 fss 6513 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
185183, 155, 184sylancl 590 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
186185adantr 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
18713adantr 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
18835adantr 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
189 xp2nd 7727 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑈) ∈ (0...𝑁))
1905, 189syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑈) ∈ (0...𝑁))
191 eldifsn 4678 . . . . . . . . . . . . . . . . 17 ((2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)))
192191biimpri 231 . . . . . . . . . . . . . . . 16 (((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
193190, 192sylan 584 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
194159, 179, 186, 187, 188, 193poimirlem2 35332 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
195194ex 417 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑈) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
196195necon1bd 2970 . . . . . . . . . . . 12 (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑈) = (2nd𝑇)))
197158, 196mpd 15 . . . . . . . . . . 11 (𝜑 → (2nd𝑈) = (2nd𝑇))
198197oveq1d 7166 . . . . . . . . . 10 (𝜑 → ((2nd𝑈) − 1) = ((2nd𝑇) − 1))
199198oveq2d 7167 . . . . . . . . 9 (𝜑 → (1...((2nd𝑈) − 1)) = (1...((2nd𝑇) − 1)))
200199eleq2d 2838 . . . . . . . 8 (𝜑 → (𝑘 ∈ (1...((2nd𝑈) − 1)) ↔ 𝑘 ∈ (1...((2nd𝑇) − 1))))
201200biimpar 482 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
20238adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑁 ∈ ℕ)
2031adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑈𝑆)
204197, 35eqeltrd 2853 . . . . . . . . 9 (𝜑 → (2nd𝑈) ∈ (1...(𝑁 − 1)))
205204adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
206 simpr 489 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
207202, 3, 203, 205, 206poimirlem6 35336 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
208201, 207syldan 595 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
20938adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑁 ∈ ℕ)
21019adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑇𝑆)
21135adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
212 simpr 489 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑇) − 1)))
213209, 3, 210, 211, 212poimirlem6 35336 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
214208, 213eqtr3d 2796 . . . . 5 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
215197oveq1d 7166 . . . . . . . . . . 11 (𝜑 → ((2nd𝑈) + 1) = ((2nd𝑇) + 1))
216215oveq1d 7166 . . . . . . . . . 10 (𝜑 → (((2nd𝑈) + 1) + 1) = (((2nd𝑇) + 1) + 1))
217216oveq1d 7166 . . . . . . . . 9 (𝜑 → ((((2nd𝑈) + 1) + 1)...𝑁) = ((((2nd𝑇) + 1) + 1)...𝑁))
218217eleq2d 2838 . . . . . . . 8 (𝜑 → (𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
219218biimpar 482 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
22038adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
2211adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑈𝑆)
222204adantr 485 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
223 simpr 489 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
224220, 3, 221, 222, 223poimirlem7 35337 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
225219, 224syldan 595 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
22638adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
22719adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑇𝑆)
22835adantr 485 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
229 simpr 489 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
230226, 3, 227, 228, 229poimirlem7 35337 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
231225, 230eqtr3d 2796 . . . . 5 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
232214, 231jaodan 956 . . . 4 ((𝜑 ∧ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
233130, 232syldan 595 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
234 fvres 6678 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
235234adantl 486 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
236 fvres 6678 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
237236adantl 486 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
238233, 235, 2373eqtr4d 2804 . 2 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘))
23918, 34, 238eqfnfvd 6797 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   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  {cab 2736   ≠ wne 2952  ∃*wrmo 3074  {crab 3075  ⦋csb 3806   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  ifcif 4421  {csn 4523  {cpr 4525   class class class wbr 5033   ↦ cmpt 5113   × cxp 5523   ↾ cres 5527   “ cima 5528   Fn wfn 6331  ⟶wf 6332  –1-1-onto→wf1o 6335  ‘cfv 6336  ℩crio 7108  (class class class)co 7151   ∘f cof 7404  1st c1st 7692  2nd c2nd 7693   ↑m cmap 8417  ℂcc 10566  ℝcr 10567  0cc0 10568  1c1 10569   + caddc 10571   < clt 10706   ≤ cle 10707   − cmin 10901  ℕcn 11667  2c2 11722  ℤcz 12013  ℤ≥cuz 12275  ...cfz 12932  ..^cfzo 13075 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7406  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-n0 11928  df-z 12014  df-uz 12276  df-fz 12933  df-fzo 13076 This theorem is referenced by:  poimirlem9  35339
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