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Theorem poimirlem8 33084
Description: Lemma for poimir 33109, establishing that away from the opposite vertex the walks in poimirlem9 33085 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 3346 . . . . . . . . 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 2716 . . . . . . . 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 7150 . . . . . . 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 7151 . . . . . 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 6163 . . . . . 6 (2nd ‘(1st𝑈)) ∈ V
11 f1oeq1 6089 . . . . . 6 (𝑓 = (2nd ‘(1st𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
1210, 11elab 3337 . . . . 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 6100 . . . 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 3720 . . 3 ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ (1...𝑁)
17 fnssres 5967 . . 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 693 . 2 (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
19 poimirlem9.1 . . . . . . . 8 (𝜑𝑇𝑆)
20 elrabi 3346 . . . . . . . . 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 2716 . . . . . . . 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 7150 . . . . . . 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 7151 . . . . . 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 6163 . . . . . 6 (2nd ‘(1st𝑇)) ∈ V
28 f1oeq1 6089 . . . . . 6 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2927, 28elab 3337 . . . . 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 6100 . . . 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 5967 . . 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 693 . 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 12344 . . . . . . . . . . . 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 10988 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℂ)
40 npcan1 10407 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
4139, 40syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4241oveq2d 6626 . . . . . . . . . . 11 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
4337, 42eleqtrd 2700 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
44 fzsplit 12317 . . . . . . . . . 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 3710 . . . . . . . 8 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
47 difundir 3861 . . . . . . . . 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 12320 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
4935, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℕ)
5049nncnd 10988 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑇) ∈ ℂ)
51 npcan1 10407 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℂ → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) − 1) + 1) = (2nd𝑇))
53 nnuz 11675 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
5449, 53syl6eleq 2708 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘1))
5552, 54eqeltrd 2698 . . . . . . . . . . . . . 14 (𝜑 → (((2nd𝑇) − 1) + 1) ∈ (ℤ‘1))
5649nnzd 11433 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd𝑇) ∈ ℤ)
57 peano2zm 11372 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℤ → ((2nd𝑇) − 1) ∈ ℤ)
5856, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℤ)
59 uzid 11654 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) − 1) ∈ ℤ → ((2nd𝑇) − 1) ∈ (ℤ‘((2nd𝑇) − 1)))
60 peano2uz 11693 . . . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)))
63 peano2uz 11693 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (ℤ‘((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
6462, 63syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑇) + 1) ∈ (ℤ‘((2nd𝑇) − 1)))
65 fzsplit2 12316 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 13 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))))
6752oveq1d 6625 . . . . . . . . . . . . . . 15 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = ((2nd𝑇)...((2nd𝑇) + 1)))
68 fzpr 12346 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ ℤ → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
6956, 68syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7067, 69eqtrd 2655 . . . . . . . . . . . . . 14 (𝜑 → ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1)) = {(2nd𝑇), ((2nd𝑇) + 1)})
7170uneq2d 3750 . . . . . . . . . . . . 13 (𝜑 → ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) − 1) + 1)...((2nd𝑇) + 1))) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7266, 71eqtrd 2655 . . . . . . . . . . . 12 (𝜑 → (1...((2nd𝑇) + 1)) = ((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}))
7372difeq1d 3710 . . . . . . . . . . 11 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
7449nnred 10987 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) ∈ ℝ)
7574ltm1d 10908 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < (2nd𝑇))
7658zred 11434 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) − 1) ∈ ℝ)
7776, 74ltnled 10136 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1)))
7875, 77mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (2nd𝑇) ≤ ((2nd𝑇) − 1))
79 elfzle2 12295 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (1...((2nd𝑇) − 1)) → (2nd𝑇) ≤ ((2nd𝑇) − 1))
8078, 79nsyl 135 . . . . . . . . . . . . . 14 (𝜑 → ¬ (2nd𝑇) ∈ (1...((2nd𝑇) − 1)))
81 difsn 4302 . . . . . . . . . . . . . 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 10161 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd𝑇) + 1) ∈ ℝ)
8474, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
8574ltp1d 10906 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
8676, 74, 84, 75, 85lttrd 10150 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) − 1) < ((2nd𝑇) + 1))
8776, 84ltnled 10136 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) − 1) < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1)))
8886, 87mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
89 elfzle2 12295 . . . . . . . . . . . . . . 15 (((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)) → ((2nd𝑇) + 1) ≤ ((2nd𝑇) − 1))
9088, 89nsyl 135 . . . . . . . . . . . . . 14 (𝜑 → ¬ ((2nd𝑇) + 1) ∈ (1...((2nd𝑇) − 1)))
91 difsn 4302 . . . . . . . . . . . . . 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 3798 . . . . . . . . . . . 12 (𝜑 → (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))))
94 difun2 4025 . . . . . . . . . . . . 13 (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
95 df-pr 4156 . . . . . . . . . . . . . 14 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
9695difeq2i 3708 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
97 difundi 3860 . . . . . . . . . . . . 13 ((1...((2nd𝑇) − 1)) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)}))
9894, 96, 973eqtrri 2648 . . . . . . . . . . . 12 (((1...((2nd𝑇) − 1)) ∖ {(2nd𝑇)}) ∩ ((1...((2nd𝑇) − 1)) ∖ {((2nd𝑇) + 1)})) = (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
99 inidm 3805 . . . . . . . . . . . 12 ((1...((2nd𝑇) − 1)) ∩ (1...((2nd𝑇) − 1))) = (1...((2nd𝑇) − 1))
10093, 98, 993eqtr3g 2678 . . . . . . . . . . 11 (𝜑 → (((1...((2nd𝑇) − 1)) ∪ {(2nd𝑇), ((2nd𝑇) + 1)}) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
10173, 100eqtrd 2655 . . . . . . . . . 10 (𝜑 → ((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (1...((2nd𝑇) − 1)))
102 peano2re 10161 . . . . . . . . . . . . . . . . 17 (((2nd𝑇) + 1) ∈ ℝ → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10384, 102syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((2nd𝑇) + 1) + 1) ∈ ℝ)
10484ltp1d 10906 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1))
10574, 84, 103, 85, 104lttrd 10150 . . . . . . . . . . . . . . 15 (𝜑 → (2nd𝑇) < (((2nd𝑇) + 1) + 1))
10674, 103ltnled 10136 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑇) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇)))
107105, 106mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
108 elfzle1 12294 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁) → (((2nd𝑇) + 1) + 1) ≤ (2nd𝑇))
109107, 108nsyl 135 . . . . . . . . . . . . 13 (𝜑 → ¬ (2nd𝑇) ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
110 difsn 4302 . . . . . . . . . . . . 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 10136 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd𝑇) + 1) < (((2nd𝑇) + 1) + 1) ↔ ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1)))
113104, 112mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ¬ (((2nd𝑇) + 1) + 1) ≤ ((2nd𝑇) + 1))
114 elfzle1 12294 . . . . . . . . . . . . . 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 4302 . . . . . . . . . . . . 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 3798 . . . . . . . . . . 11 (𝜑 → ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)))
11995difeq2i 3708 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
120 difundi 3860 . . . . . . . . . . . 12 (((((2nd𝑇) + 1) + 1)...𝑁) ∖ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)}))
121119, 120eqtr2i 2644 . . . . . . . . . . 11 ((((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇)}) ∩ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {((2nd𝑇) + 1)})) = (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
122 inidm 3805 . . . . . . . . . . 11 (((((2nd𝑇) + 1) + 1)...𝑁) ∩ ((((2nd𝑇) + 1) + 1)...𝑁)) = ((((2nd𝑇) + 1) + 1)...𝑁)
123118, 121, 1223eqtr3g 2678 . . . . . . . . . 10 (𝜑 → (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((((2nd𝑇) + 1) + 1)...𝑁))
124101, 123uneq12d 3751 . . . . . . . . 9 (𝜑 → (((1...((2nd𝑇) + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (((((2nd𝑇) + 1) + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12547, 124syl5eq 2667 . . . . . . . 8 (𝜑 → (((1...((2nd𝑇) + 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
12646, 125eqtrd 2655 . . . . . . 7 (𝜑 → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁)))
127126eleq2d 2684 . . . . . 6 (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2nd𝑇) − 1)) ∪ ((((2nd𝑇) + 1) + 1)...𝑁))))
128 elun 3736 . . . . . 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 501 . . . 4 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
131 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
132131breq2d 4630 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
133132ifbid 4085 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
134133csbeq1d 3525 . . . . . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
136135fveq2d 6157 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
137135fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
138137imaeq1d 5429 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
139138xpeq1d 5103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
140137imaeq1d 5429 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
141140xpeq1d 5103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
142139, 141uneq12d 3751 . . . . . . . . . . . . . . . . . . . . 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 6628 . . . . . . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . . 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 4710 . . . . . . . . . . . . . . . . 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 2631 . . . . . . . . . . . . . . . 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 3352 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 7150 . . . . . . . . . . . . . . . 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 7831 . . . . . . . . . . . . . . 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 12419 . . . . . . . . . . . . . . 15 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
156155ssriv 3591 . . . . . . . . . . . . . 14 (0..^𝐾) ⊆ ℤ
157 fss 6018 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
158154, 156, 157sylancl 693 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
15938, 150, 158, 30, 35poimirlem1 33077 . . . . . . . . . . . 12 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
16038adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
161 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (2nd𝑡) = (2nd𝑈))
162161breq2d 4630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑈)))
163162ifbid 4085 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑈 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)))
164163csbeq1d 3525 . . . . . . . . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (1st𝑡) = (1st𝑈))
166165fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑈 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑈)))
167165fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑈 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑈)))
168167imaeq1d 5429 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑗)))
169168xpeq1d 5103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}))
170167imaeq1d 5429 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)))
171170xpeq1d 5103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
172169, 171uneq12d 3751 . . . . . . . . . . . . . . . . . . . . . . . 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 6628 . . . . . . . . . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . . . . . 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 4710 . . . . . . . . . . . . . . . . . . . 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 2631 . . . . . . . . . . . . . . . . . . 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 3352 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 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})))))
182 xp1st 7150 . . . . . . . . . . . . . . . . . . 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 7831 . . . . . . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
187185, 156, 186sylancl 693 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
188187adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (1st ‘(1st𝑈)):(1...𝑁)⟶ℤ)
18913adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
19035adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
191 xp2nd 7151 . . . . . . . . . . . . . . . . 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 4292 . . . . . . . . . . . . . . . . 17 ((2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)))
194193biimpri 218 . . . . . . . . . . . . . . . 16 (((2nd𝑈) ∈ (0...𝑁) ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
195192, 194sylan 488 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → (2nd𝑈) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
196160, 181, 188, 189, 190, 195poimirlem2 33078 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑈) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
197196ex 450 . . . . . . . . . . . . 13 (𝜑 → ((2nd𝑈) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
198197necon1bd 2808 . . . . . . . . . . . 12 (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑈) = (2nd𝑇)))
199159, 198mpd 15 . . . . . . . . . . 11 (𝜑 → (2nd𝑈) = (2nd𝑇))
200199oveq1d 6625 . . . . . . . . . 10 (𝜑 → ((2nd𝑈) − 1) = ((2nd𝑇) − 1))
201200oveq2d 6626 . . . . . . . . 9 (𝜑 → (1...((2nd𝑈) − 1)) = (1...((2nd𝑇) − 1)))
202201eleq2d 2684 . . . . . . . 8 (𝜑 → (𝑘 ∈ (1...((2nd𝑈) − 1)) ↔ 𝑘 ∈ (1...((2nd𝑇) − 1))))
203202biimpar 502 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
20438adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑁 ∈ ℕ)
2051adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑈𝑆)
206199, 35eqeltrd 2698 . . . . . . . . 9 (𝜑 → (2nd𝑈) ∈ (1...(𝑁 − 1)))
207206adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
208 simpr 477 . . . . . . . 8 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → 𝑘 ∈ (1...((2nd𝑈) − 1)))
209204, 3, 205, 207, 208poimirlem6 33082 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑈) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
210203, 209syldan 487 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
21138adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑁 ∈ ℕ)
21219adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑇𝑆)
21335adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
214 simpr 477 . . . . . . 7 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → 𝑘 ∈ (1...((2nd𝑇) − 1)))
215211, 3, 212, 213, 214poimirlem6 33082 . . . . . 6 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹𝑘)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
216210, 215eqtr3d 2657 . . . . 5 ((𝜑𝑘 ∈ (1...((2nd𝑇) − 1))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
217199oveq1d 6625 . . . . . . . . . . 11 (𝜑 → ((2nd𝑈) + 1) = ((2nd𝑇) + 1))
218217oveq1d 6625 . . . . . . . . . 10 (𝜑 → (((2nd𝑈) + 1) + 1) = (((2nd𝑇) + 1) + 1))
219218oveq1d 6625 . . . . . . . . 9 (𝜑 → ((((2nd𝑈) + 1) + 1)...𝑁) = ((((2nd𝑇) + 1) + 1)...𝑁))
220219eleq2d 2684 . . . . . . . 8 (𝜑 → (𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)))
221220biimpar 502 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
22238adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
2231adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑈𝑆)
224206adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (2nd𝑈) ∈ (1...(𝑁 − 1)))
225 simpr 477 . . . . . . . 8 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁))
226222, 3, 223, 224, 225poimirlem7 33083 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑈) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
227221, 226syldan 487 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑈))‘𝑘))
22838adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
22919adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑇𝑆)
23035adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
231 simpr 477 . . . . . . 7 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))
232228, 3, 229, 230, 231poimirlem7 33083 . . . . . 6 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑘))
233227, 232eqtr3d 2657 . . . . 5 ((𝜑𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁)) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
234216, 233jaodan 825 . . . 4 ((𝜑 ∧ (𝑘 ∈ (1...((2nd𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
235130, 234syldan 487 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ((2nd ‘(1st𝑈))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
236 fvres 6169 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
237236adantl 482 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑈))‘𝑘))
238 fvres 6169 . . . 4 (𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
239238adantl 482 . . 3 ((𝜑𝑘 ∈ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → (((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st𝑇))‘𝑘))
240235, 237, 2393eqtr4d 2665 . 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 6275 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 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  ∃*wrmo 2910  {crab 2911  csb 3518  cdif 3556  cun 3557  cin 3558  wss 3559  ifcif 4063  {csn 4153  {cpr 4155   class class class wbr 4618  cmpt 4678   × cxp 5077  cres 5081  cima 5082   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  crio 6570  (class class class)co 6610  𝑓 cof 6855  1st c1st 7118  2nd c2nd 7119  𝑚 cmap 7809  cc 9886  cr 9887  0cc0 9888  1c1 9889   + caddc 9891   < clt 10026  cle 10027  cmin 10218  cn 10972  2c2 11022  cz 11329  cuz 11639  ...cfz 12276  ..^cfzo 12414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415
This theorem is referenced by:  poimirlem9  33085
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