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Theorem poimirlem10 33775
Description: Lemma for poimir 33798 establishing the cube that yields the simplex that yields a face if the opposite vertex was first on the walk. (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}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
poimirlem12.2 (𝜑𝑇𝑆)
poimirlem11.3 (𝜑 → (2nd𝑇) = 0)
Assertion
Ref Expression
poimirlem10 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑆,𝑗,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem10
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ovexd 6876 . 2 (𝜑 → (1...𝑁) ∈ V)
2 poimirlem22.1 . . . 4 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
3 poimir.0 . . . . . 6 (𝜑𝑁 ∈ ℕ)
4 nnm1nn0 11581 . . . . . 6 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
53, 4syl 17 . . . . 5 (𝜑 → (𝑁 − 1) ∈ ℕ0)
6 nn0fz0 12645 . . . . 5 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
75, 6sylib 209 . . . 4 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
82, 7ffvelrnd 6550 . . 3 (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑𝑚 (1...𝑁)))
9 elmapfn 8083 . . 3 ((𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑𝑚 (1...𝑁)) → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
108, 9syl 17 . 2 (𝜑 → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
11 1ex 10289 . . 3 1 ∈ V
12 fnconstg 6275 . . 3 (1 ∈ V → ((1...𝑁) × {1}) Fn (1...𝑁))
1311, 12mp1i 13 . 2 (𝜑 → ((1...𝑁) × {1}) Fn (1...𝑁))
14 poimirlem12.2 . . . . . 6 (𝜑𝑇𝑆)
15 elrabi 3514 . . . . . . 7 (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)))
16 poimirlem22.s . . . . . . 7 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}))))}
1715, 16eleq2s 2862 . . . . . 6 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
1814, 17syl 17 . . . . 5 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
19 xp1st 7398 . . . . 5 (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)}))
2018, 19syl 17 . . . 4 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21 xp1st 7398 . . . 4 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
2220, 21syl 17 . . 3 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
23 elmapfn 8083 . . 3 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
2422, 23syl 17 . 2 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
25 fveq2 6375 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
2625breq2d 4821 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
2726ifbid 4265 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
2827csbeq1d 3698 . . . . . . . . . . . 12 (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}))))
29 2fveq3 6380 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
30 2fveq3 6380 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
3130imaeq1d 5647 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
3231xpeq1d 5306 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
3330imaeq1d 5647 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
3433xpeq1d 5306 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
3532, 34uneq12d 3930 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
3629, 35oveq12d 6860 . . . . . . . . . . . . 13 (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}))))
3736csbeq2dv 4153 . . . . . . . . . . . 12 (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}))))
3828, 37eqtrd 2799 . . . . . . . . . . 11 (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}))))
3938mpteq2dv 4904 . . . . . . . . . 10 (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})))))
4039eqeq2d 2775 . . . . . . . . 9 (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}))))))
4140, 16elrab2 3523 . . . . . . . 8 (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}))))))
4241simprbi 490 . . . . . . 7 (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})))))
4314, 42syl 17 . . . . . 6 (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})))))
44 poimirlem11.3 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) = 0)
45 breq12 4814 . . . . . . . . . . . 12 ((𝑦 = (𝑁 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4644, 45sylan2 586 . . . . . . . . . . 11 ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4746ancoms 450 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
48 oveq1 6849 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
493nncnd 11292 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
50 npcan1 10709 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5149, 50syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
5248, 51sylan9eqr 2821 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
5347, 52ifbieq2d 4268 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
545nn0ge0d 11601 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑁 − 1))
55 0red 10297 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℝ)
565nn0red 11599 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5755, 56lenltd 10437 . . . . . . . . . . . 12 (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
5854, 57mpbid 223 . . . . . . . . . . 11 (𝜑 → ¬ (𝑁 − 1) < 0)
5958iffalsed 4254 . . . . . . . . . 10 (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6059adantr 472 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6153, 60eqtrd 2799 . . . . . . . 8 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
6261csbeq1d 3698 . . . . . . 7 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑁 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
63 oveq2 6850 . . . . . . . . . . . . . . 15 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
6463imaeq2d 5648 . . . . . . . . . . . . . 14 (𝑗 = 𝑁 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
65 xp2nd 7399 . . . . . . . . . . . . . . . . 17 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
6620, 65syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
67 fvex 6388 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
68 f1oeq1 6310 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6967, 68elab 3504 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
7066, 69sylib 209 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1ofo 6327 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
72 foima 6303 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
7370, 71, 723syl 18 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
7464, 73sylan9eqr 2821 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = (1...𝑁))
7574xpeq1d 5306 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76 oveq1 6849 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
7776oveq1d 6857 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
783nnred 11291 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
7978ltp1d 11208 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 < (𝑁 + 1))
803nnzd 11728 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
8180peano2zd 11732 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 + 1) ∈ ℤ)
82 fzn 12564 . . . . . . . . . . . . . . . . . 18 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8381, 80, 82syl2anc 579 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8479, 83mpbid 223 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
8577, 84sylan9eqr 2821 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
8685imaeq2d 5648 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ∅))
8786xpeq1d 5306 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ∅) × {0}))
88 ima0 5663 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
8988xpeq1i 5303 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = (∅ × {0})
90 0xp 5369 . . . . . . . . . . . . . 14 (∅ × {0}) = ∅
9189, 90eqtri 2787 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = ∅
9287, 91syl6eq 2815 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
9375, 92uneq12d 3930 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94 un0 4129 . . . . . . . . . . 11 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
9593, 94syl6eq 2815 . . . . . . . . . 10 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
9695oveq2d 6858 . . . . . . . . 9 ((𝜑𝑗 = 𝑁) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
973, 96csbied 3718 . . . . . . . 8 (𝜑𝑁 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
9897adantr 472 . . . . . . 7 ((𝜑𝑦 = (𝑁 − 1)) → 𝑁 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
9962, 98eqtrd 2799 . . . . . 6 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
100 ovexd 6876 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})) ∈ V)
10143, 99, 7, 100fvmptd 6477 . . . . 5 (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
102101fveq1d 6377 . . . 4 (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
103102adantr 472 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
104 inidm 3982 . . . 4 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105 eqidd 2766 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
10611fvconst2 6662 . . . . 5 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107106adantl 473 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
10824, 13, 1, 1, 104, 105, 107ofval 7104 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
109103, 108eqtrd 2799 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
110 elmapi 8082 . . . . . . 7 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
11122, 110syl 17 . . . . . 6 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
112111ffvelrnda 6549 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
113 elfzonn0 12721 . . . . 5 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
114112, 113syl 17 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
115114nn0cnd 11600 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
116 pncan1 10708 . . 3 (((1st ‘(1st𝑇))‘𝑛) ∈ ℂ → ((((1st ‘(1st𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st𝑇))‘𝑛))
117115, 116syl 17 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st𝑇))‘𝑛))
1181, 10, 13, 24, 109, 107, 117offveq 7116 1 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  {crab 3059  Vcvv 3350  csb 3691  cun 3730  c0 4079  ifcif 4243  {csn 4334   class class class wbr 4809  cmpt 4888   × cxp 5275  cima 5280   Fn wfn 6063  wf 6064  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  𝑓 cof 7093  1st c1st 7364  2nd c2nd 7365  𝑚 cmap 8060  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cle 10329  cmin 10520  cn 11274  0cn0 11538  cz 11624  ...cfz 12533  ..^cfzo 12673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674
This theorem is referenced by:  poimirlem11  33776  poimirlem13  33778
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