Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem10 Structured version   Visualization version   GIF version

Theorem poimirlem10 34379
Description: Lemma for poimir 34402 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 7041 . 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 11775 . . . . . 6 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
53, 4syl 17 . . . . 5 (𝜑 → (𝑁 − 1) ∈ ℕ0)
6 nn0fz0 12844 . . . . 5 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
75, 6sylib 219 . . . 4 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
82, 7ffvelrnd 6708 . . 3 (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑𝑚 (1...𝑁)))
9 elmapfn 8270 . . 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 10472 . . 3 1 ∈ V
12 fnconstg 6427 . . 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 3608 . . . . . . 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 2899 . . . . . 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 7568 . . . . 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 7568 . . . 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 8270 . . 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 6530 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
2625breq2d 4968 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
2726ifbid 4397 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
2827csbeq1d 3810 . . . . . . . . . . . 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 6535 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
30 2fveq3 6535 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
3130imaeq1d 5797 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
3231xpeq1d 5464 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
3330imaeq1d 5797 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
3433xpeq1d 5464 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
3532, 34uneq12d 4056 . . . . . . . . . . . . . 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 7025 . . . . . . . . . . . . 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 3813 . . . . . . . . . . . 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 2829 . . . . . . . . . . 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 5050 . . . . . . . . . 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 2803 . . . . . . . . 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 3616 . . . . . . . 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 497 . . . . . . 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 4961 . . . . . . . . . . . 12 ((𝑦 = (𝑁 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4644, 45sylan2 592 . . . . . . . . . . 11 ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4746ancoms 459 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
48 oveq1 7014 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
493nncnd 11491 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
50 npcan1 10902 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5149, 50syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
5248, 51sylan9eqr 2851 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
5347, 52ifbieq2d 4400 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
545nn0ge0d 11795 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑁 − 1))
55 0red 10479 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℝ)
565nn0red 11793 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5755, 56lenltd 10622 . . . . . . . . . . . 12 (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
5854, 57mpbid 233 . . . . . . . . . . 11 (𝜑 → ¬ (𝑁 − 1) < 0)
5958iffalsed 4386 . . . . . . . . . 10 (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6059adantr 481 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6153, 60eqtrd 2829 . . . . . . . 8 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
6261csbeq1d 3810 . . . . . . 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 7015 . . . . . . . . . . . . . . 15 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
6463imaeq2d 5798 . . . . . . . . . . . . . 14 (𝑗 = 𝑁 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
65 xp2nd 7569 . . . . . . . . . . . . . . . . 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 6543 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
68 f1oeq1 6464 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6967, 68elab 3600 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
7066, 69sylib 219 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1ofo 6482 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
72 foima 6455 . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = (1...𝑁))
7574xpeq1d 5464 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76 oveq1 7014 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
7776oveq1d 7022 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
783nnred 11490 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
7978ltp1d 11407 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 < (𝑁 + 1))
803nnzd 11924 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
8180peano2zd 11928 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 + 1) ∈ ℤ)
82 fzn 12762 . . . . . . . . . . . . . . . . . 18 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8381, 80, 82syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8479, 83mpbid 233 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
8577, 84sylan9eqr 2851 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
8685imaeq2d 5798 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ∅))
8786xpeq1d 5464 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ∅) × {0}))
88 ima0 5813 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
8988xpeq1i 5461 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = (∅ × {0})
90 0xp 5527 . . . . . . . . . . . . . 14 (∅ × {0}) = ∅
9189, 90eqtri 2817 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = ∅
9287, 91syl6eq 2845 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
9375, 92uneq12d 4056 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94 un0 4258 . . . . . . . . . . 11 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
9593, 94syl6eq 2845 . . . . . . . . . 10 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
9695oveq2d 7023 . . . . . . . . 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 3839 . . . . . . . 8 (𝜑𝑁 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
9897adantr 481 . . . . . . 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 2829 . . . . . 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 7041 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})) ∈ V)
10143, 99, 7, 100fvmptd 6632 . . . . 5 (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1})))
102101fveq1d 6532 . . . 4 (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
103102adantr 481 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛))
104 inidm 4110 . . . 4 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105 eqidd 2794 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
10611fvconst2 6824 . . . . 5 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107106adantl 482 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
10824, 13, 1, 1, 104, 105, 107ofval 7267 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
109103, 108eqtrd 2829 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
110 elmapi 8269 . . . . . . 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 6707 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
113 elfzonn0 12920 . . . . 5 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
114112, 113syl 17 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
115114nn0cnd 11794 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
116 pncan1 10901 . . 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 7279 1 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1520  wcel 2079  {cab 2773  {crab 3107  Vcvv 3432  csb 3806  cun 3852  c0 4206  ifcif 4375  {csn 4466   class class class wbr 4956  cmpt 5035   × cxp 5433  cima 5438   Fn wfn 6212  wf 6213  ontowfo 6215  1-1-ontowf1o 6216  cfv 6217  (class class class)co 7007  𝑓 cof 7256  1st c1st 7534  2nd c2nd 7535  𝑚 cmap 8247  cc 10370  0cc0 10372  1c1 10373   + caddc 10375   < clt 10510  cle 10511  cmin 10706  cn 11475  0cn0 11734  cz 11818  ...cfz 12731  ..^cfzo 12872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-of 7258  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-er 8130  df-map 8249  df-en 8348  df-dom 8349  df-sdom 8350  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-n0 11735  df-z 11819  df-uz 12083  df-fz 12732  df-fzo 12873
This theorem is referenced by:  poimirlem11  34380  poimirlem13  34382
  Copyright terms: Public domain W3C validator