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Theorem poimirlem8 34786
Description: Lemma for poimir 34811, establishing that away from the opposite vertex the walks in poimirlem9 34787 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 3679 . . . . . . . . 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 2936 . . . . . . . 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 7717 . . . . . . 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 7718 . . . . . 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 6682 . . . . . 6 (2nd ‘(1st𝑈)) ∈ V
11 f1oeq1 6603 . . . . . 6 (𝑓 = (2nd ‘(1st𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3671 . . . . 5 ((2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
139, 12sylib 219 . . . 4 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14 f1ofn 6615 . . . 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 4112 . . 3 ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)
17 fnssres 6469 . . 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 586 . 2 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
19 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
20 elrabi 3679 . . . . . . . . 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 2936 . . . . . . . 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 7717 . . . . . . 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 7718 . . . . . 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 6682 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
28 f1oeq1 6603 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2927, 28elab 3671 . . . . 5 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
3026, 29sylib 219 . . . 4 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31 f1ofn 6615 . . . 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 6469 . . 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 586 . 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 12955 . . . . . . . . . . . 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 11648 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℂ)
40 npcan1 11059 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
4139, 40syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4241oveq2d 7166 . . . . . . . . . . 11 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
4337, 42eleqtrd 2920 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
44 fzsplit 12928 . . . . . . . . . 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 4102 . . . . . . . 8 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
47 difundir 4261 . . . . . . . . 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 12931 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
4935, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℕ)
5049nncnd 11648 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑇) ∈ ℂ)
51 npcan1 11059 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
53 nnuz 12275 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
5449, 53syl6eleq 2928 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘1))
5552, 54eqeltrd 2918 . . . . . . . . . . . . . 14 (𝜑 → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘1))
5649nnzd 12080 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd𝑇) ∈ ℤ)
57 peano2zm 12019 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
5856, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
59 uzid 12252 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) − 1) ∈ ℤ → ((2nd𝑇) − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
60 peano2uz 12295 . . . . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)))
63 peano2uz 12295 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6462, 63syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
65 fzsplit2 12927 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 13 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))))
6752oveq1d 7165 . . . . . . . . . . . . . . 15 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = ((2nd𝑇)...((2nd𝑇) + 1)))
68 fzpr 12957 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℤ → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
6956, 68syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7067, 69eqtrd 2861 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7170uneq2d 4143 . . . . . . . . . . . . 13 (𝜑 → ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7266, 71eqtrd 2861 . . . . . . . . . . . 12 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7372difeq1d 4102 . . . . . . . . . . 11 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
7449nnred 11647 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
7574ltm1d 11566 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
7658zred 12081 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
7776, 74ltnled 10781 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1)))
7875, 77mpbid 233 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1))
79 elfzle2 12906 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (1...((2nd𝑇) − 1)) → (2nd𝑇) ≤ ((2nd𝑇) − 1))
8078, 79nsyl 142 . . . . . . . . . . . . . 14 (𝜑 → ¬ (2nd𝑇) ∈ (1...((2nd𝑇) − 1)))
81 difsn 4730 . . . . . . . . . . . . . 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 10807 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd𝑇) + 1) ∈ ℝ)
8474, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
8574ltp1d 11564 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
8676, 74, 84, 75, 85lttrd 10795 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < ((2nd𝑇) + 1))
8776, 84ltnled 10781 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1)))
8886, 87mpbid 233 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
89 elfzle2 12906 . . . . . . . . . . . . . . 15 (((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
9088, 89nsyl 142 . . . . . . . . . . . . . 14 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)))
91 difsn 4730 . . . . . . . . . . . . . 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 4194 . . . . . . . . . . . 12 (𝜑 → (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))))
94 difun2 4432 . . . . . . . . . . . . 13 (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
95 df-pr 4567 . . . . . . . . . . . . . 14 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
9695difeq2i 4100 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
97 difundi 4260 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)}))
9894, 96, 973eqtrri 2854 . . . . . . . . . . . 12 (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
99 inidm 4199 . . . . . . . . . . . 12 ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))) = (1...((2nd𝑇) − 1))
10093, 98, 993eqtr3g 2884 . . . . . . . . . . 11 (𝜑 → (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
10173, 100eqtrd 2861 . . . . . . . . . 10 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
102 peano2re 10807 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) + 1) ∈ ℝ → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10384, 102syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10484ltp1d 11564 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
10574, 84, 103, 85, 104lttrd 10795 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) < (((2nd𝑇) + 1) + 1))
10674, 103ltnled 10781 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇)))
107105, 106mpbid 233 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
108 elfzle1 12905 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
109107, 108nsyl 142 . . . . . . . . . . . . 13 (𝜑 → ¬ (2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
110 difsn 4730 . . . . . . . . . . . . 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 10781 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1)))
113104, 112mpbid 233 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
114 elfzle1 12905 . . . . . . . . . . . . . 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 4730 . . . . . . . . . . . . 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 4194 . . . . . . . . . . 11 (𝜑 → ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)))
11995difeq2i 4100 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
120 difundi 4260 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)}))
121119, 120eqtr2i 2850 . . . . . . . . . . 11 ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
122 inidm 4199 . . . . . . . . . . 11 (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)) = ((((2nd𝑇) + 1) + 1)...𝑁)
123118, 121, 1223eqtr3g 2884 . . . . . . . . . 10 (𝜑 → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
124101, 123uneq12d 4144 . . . . . . . . 9 (𝜑 → (((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12547, 124syl5eq 2873 . . . . . . . 8 (𝜑 → (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12646, 125eqtrd 2861 . . . . . . 7 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
127126eleq2d 2903 . . . . . 6 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁))))
128 elun 4129 . . . . . 6 (𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
129127, 128syl6bb 288 . . . . 5 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))))
130129biimpa 477 . . . 4 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
131 fveq2 6669 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
132131breq2d 5075 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
133132ifbid 4492 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
134133csbeq1d 3891 . . . . . . . . . . . . . . . . . . 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 6674 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
136 2fveq3 6674 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
137136imaeq1d 5927 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
138137xpeq1d 5583 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
139136imaeq1d 5927 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
140139xpeq1d 5583 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
141138, 140uneq12d 4144 . . . . . . . . . . . . . . . . . . . . 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 7168 . . . . . . . . . . . . . . . . . . . 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 3894 . . . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . . 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 3687 . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . 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 7717 . . . . . . . . . . . . . . . 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 8423 . . . . . . . . . . . . . . 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 13033 . . . . . . . . . . . . . . 15 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155154ssriv 3975 . . . . . . . . . . . . . 14 (0..^𝐾) ⊆ ℤ
156 fss 6526 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
157153, 155, 156sylancl 586 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
15838, 149, 157, 30, 35poimirlem1 34779 . . . . . . . . . . . 12 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
15938adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
160 fveq2 6669 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (2nd𝑡) = (2nd𝑈))
161160breq2d 5075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑈)))
162161ifbid 4492 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑈 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)))
163162csbeq1d 3891 . . . . . . . . . . . . . . . . . . . . . 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 6674 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑈)))
165 2fveq3 6674 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑈 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑈)))
166165imaeq1d 5927 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑗)))
167166xpeq1d 5583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}))
168165imaeq1d 5927 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)))
169168xpeq1d 5583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
170167, 169uneq12d 4144 . . . . . . . . . . . . . . . . . . . . . . . 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 7168 . . . . . . . . . . . . . . . . . . . . . . 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 3894 . . . . . . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . . . . . 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 3687 . . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 7717 . . . . . . . . . . . . . . . . . . 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 8423 . . . . . . . . . . . . . . . . . 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 6526 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
185183, 155, 184sylancl 586 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
186185adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
18713adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
18835adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
189 xp2nd 7718 . . . . . . . . . . . . . . . . 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 4718 . . . . . . . . . . . . . . . . 17 ((2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)))
192191biimpri 229 . . . . . . . . . . . . . . . 16 (((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
193190, 192sylan 580 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
194159, 179, 186, 187, 188, 193poimirlem2 34780 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
195194ex 413 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑈) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
196195necon1bd 3039 . . . . . . . . . . . 12 (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑈) = (2nd𝑇)))
197158, 196mpd 15 . . . . . . . . . . 11 (𝜑 → (2nd𝑈) = (2nd𝑇))
198197oveq1d 7165 . . . . . . . . . 10 (𝜑 → ((2nd𝑈) − 1) = ((2nd𝑇) − 1))
199198oveq2d 7166 . . . . . . . . 9 (𝜑 → (1...((2nd𝑈) − 1)) = (1...((2nd𝑇) − 1)))
200199eleq2d 2903 . . . . . . . 8 (𝜑 → (𝑘 ∈ (1...((2nd𝑈) − 1)) ↔ 𝑘 ∈ (1...((2nd𝑇) − 1))))
201200biimpar 478 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
20238adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑁 ∈ ℕ)
2031adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑈𝑆)
204197, 35eqeltrd 2918 . . . . . . . . 9 (𝜑 → (2nd𝑈) ∈ (1...(𝑁 − 1)))
205204adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
206 simpr 485 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
207202, 3, 203, 205, 206poimirlem6 34784 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
208201, 207syldan 591 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
20938adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑁 ∈ ℕ)
21019adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑇𝑆)
21135adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
212 simpr 485 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑇) − 1)))
213209, 3, 210, 211, 212poimirlem6 34784 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
214208, 213eqtr3d 2863 . . . . 5 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
215197oveq1d 7165 . . . . . . . . . . 11 (𝜑 → ((2nd𝑈) + 1) = ((2nd𝑇) + 1))
216215oveq1d 7165 . . . . . . . . . 10 (𝜑 → (((2nd𝑈) + 1) + 1) = (((2nd𝑇) + 1) + 1))
217216oveq1d 7165 . . . . . . . . 9 (𝜑 → ((((2nd𝑈) + 1) + 1)...𝑁) = ((((2nd𝑇) + 1) + 1)...𝑁))
218217eleq2d 2903 . . . . . . . 8 (𝜑 → (𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
219218biimpar 478 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
22038adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
2211adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑈𝑆)
222204adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
223 simpr 485 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
224220, 3, 221, 222, 223poimirlem7 34785 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
225219, 224syldan 591 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
22638adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
22719adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑇𝑆)
22835adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
229 simpr 485 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
230226, 3, 227, 228, 229poimirlem7 34785 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
231225, 230eqtr3d 2863 . . . . 5 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
232214, 231jaodan 953 . . . 4 ((𝜑 ∧ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
233130, 232syldan 591 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
234 fvres 6688 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
235234adantl 482 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
236 fvres 6688 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
237236adantl 482 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
238233, 235, 2373eqtr4d 2871 . 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 6803 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 396  wo 843   = wceq 1530  wcel 2107  {cab 2804  wne 3021  ∃*wrmo 3146  {crab 3147  csb 3887  cdif 3937  cun 3938  cin 3939  wss 3940  ifcif 4470  {csn 4564  {cpr 4566   class class class wbr 5063  cmpt 5143   × cxp 5552  cres 5556  cima 5557   Fn wfn 6349  wf 6350  1-1-ontowf1o 6353  cfv 6354  crio 7107  (class class class)co 7150  f cof 7401  1st c1st 7683  2nd c2nd 7684  m cmap 8401  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669  cle 10670  cmin 10864  cn 11632  2c2 11686  cz 11975  cuz 12237  ...cfz 12887  ..^cfzo 13028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029
This theorem is referenced by:  poimirlem9  34787
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