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Theorem poimirlem10 33745
Description: Lemma for poimir 33768 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 6823 . 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 11534 . . . . . 6 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
53, 4syl 17 . . . . 5 (𝜑 → (𝑁 − 1) ∈ ℕ0)
6 nn0fz0 12638 . . . . 5 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
75, 6sylib 208 . . . 4 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
82, 7ffvelrnd 6501 . . 3 (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑𝑚 (1...𝑁)))
9 elmapfn 8030 . . 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 10235 . . 3 1 ∈ V
12 fnconstg 6231 . . 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 3510 . . . . . . 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 2868 . . . . . 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 7345 . . . . 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 7345 . . . 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 8030 . . 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 6330 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
2625breq2d 4798 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
2726ifbid 4247 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
2827csbeq1d 3689 . . . . . . . . . . . 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 fveq2 6330 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
3029fveq2d 6334 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
3129fveq2d 6334 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
3231imaeq1d 5604 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
3332xpeq1d 5277 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
3431imaeq1d 5604 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
3534xpeq1d 5277 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
3633, 35uneq12d 3919 . . . . . . . . . . . . . 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})))
3730, 36oveq12d 6809 . . . . . . . . . . . . 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}))))
3837csbeq2dv 4136 . . . . . . . . . . . 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}))))
3928, 38eqtrd 2805 . . . . . . . . . . 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}))))
4039mpteq2dv 4879 . . . . . . . . . 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})))))
4140eqeq2d 2781 . . . . . . . . 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}))))))
4241, 16elrab2 3518 . . . . . . . 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}))))))
4342simprbi 484 . . . . . . 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})))))
4414, 43syl 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})))))
45 poimirlem11.3 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) = 0)
46 breq12 4791 . . . . . . . . . . . 12 ((𝑦 = (𝑁 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4745, 46sylan2 580 . . . . . . . . . . 11 ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4847ancoms 446 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
49 oveq1 6798 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
503nncnd 11236 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
51 npcan1 10655 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5250, 51syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
5349, 52sylan9eqr 2827 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
5448, 53ifbieq2d 4250 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
555nn0ge0d 11554 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑁 − 1))
56 0red 10241 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℝ)
575nn0red 11552 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5856, 57lenltd 10383 . . . . . . . . . . . 12 (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
5955, 58mpbid 222 . . . . . . . . . . 11 (𝜑 → ¬ (𝑁 − 1) < 0)
6059iffalsed 4236 . . . . . . . . . 10 (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6160adantr 466 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6254, 61eqtrd 2805 . . . . . . . 8 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
6362csbeq1d 3689 . . . . . . 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}))))
64 oveq2 6799 . . . . . . . . . . . . . . 15 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
6564imaeq2d 5605 . . . . . . . . . . . . . 14 (𝑗 = 𝑁 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
66 xp2nd 7346 . . . . . . . . . . . . . . . . 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)})
6720, 66syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
68 fvex 6340 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
69 f1oeq1 6266 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
7068, 69elab 3501 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
7167, 70sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
72 f1ofo 6283 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
73 foima 6259 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
7471, 72, 733syl 18 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
7565, 74sylan9eqr 2827 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = (1...𝑁))
7675xpeq1d 5277 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
77 oveq1 6798 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
7877oveq1d 6806 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
793nnred 11235 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
8079ltp1d 11154 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 < (𝑁 + 1))
813nnzd 11681 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
8281peano2zd 11685 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 + 1) ∈ ℤ)
83 fzn 12557 . . . . . . . . . . . . . . . . . 18 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8482, 81, 83syl2anc 573 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8580, 84mpbid 222 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
8678, 85sylan9eqr 2827 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
8786imaeq2d 5605 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ∅))
8887xpeq1d 5277 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ∅) × {0}))
89 ima0 5620 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
9089xpeq1i 5274 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = (∅ × {0})
91 0xp 5337 . . . . . . . . . . . . . 14 (∅ × {0}) = ∅
9290, 91eqtri 2793 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = ∅
9388, 92syl6eq 2821 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
9476, 93uneq12d 3919 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
95 un0 4111 . . . . . . . . . . 11 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
9694, 95syl6eq 2821 . . . . . . . . . 10 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
9796oveq2d 6807 . . . . . . . . 9 ((𝜑𝑗 = 𝑁) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
983, 97csbied 3709 . . . . . . . 8 (𝜑𝑁 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
9998adantr 466 . . . . . . 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})))
10063, 99eqtrd 2805 . . . . . 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})))
101 ovexd 6823 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})) ∈ V)
10244, 100, 7, 101fvmptd 6428 . . . . 5 (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
103102fveq1d 6332 . . . 4 (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
104103adantr 466 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
105 inidm 3971 . . . 4 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
106 eqidd 2772 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
10711fvconst2 6611 . . . . 5 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
108107adantl 467 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
10924, 13, 1, 1, 105, 106, 108ofval 7051 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
110104, 109eqtrd 2805 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
111 elmapi 8029 . . . . . . 7 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
11222, 111syl 17 . . . . . 6 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
113112ffvelrnda 6500 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
114 elfzonn0 12714 . . . . 5 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
115113, 114syl 17 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
116115nn0cnd 11553 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
117 pncan1 10654 . . 3 (((1st ‘(1st𝑇))‘𝑛) ∈ ℂ → ((((1st ‘(1st𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st𝑇))‘𝑛))
118116, 117syl 17 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st𝑇))‘𝑛))
1191, 10, 13, 24, 110, 108, 118offveq 7063 1 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  Vcvv 3351  csb 3682  cun 3721  c0 4063  ifcif 4225  {csn 4316   class class class wbr 4786  cmpt 4863   × cxp 5247  cima 5252   Fn wfn 6024  wf 6025  ontowfo 6027  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  𝑓 cof 7040  1st c1st 7311  2nd c2nd 7312  𝑚 cmap 8007  cc 10134  0cc0 10136  1c1 10137   + caddc 10139   < clt 10274  cle 10275  cmin 10466  cn 11220  0cn0 11492  cz 11577  ...cfz 12526  ..^cfzo 12666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-n0 11493  df-z 11578  df-uz 11887  df-fz 12527  df-fzo 12667
This theorem is referenced by:  poimirlem11  33746  poimirlem13  33748
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