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Theorem poimirlem10 35086
 Description: Lemma for poimir 35109 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 7171 . 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 11929 . . . . . 6 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
53, 4syl 17 . . . . 5 (𝜑 → (𝑁 − 1) ∈ ℕ0)
6 nn0fz0 13003 . . . . 5 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
75, 6sylib 221 . . . 4 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
82, 7ffvelrnd 6830 . . 3 (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9 elmapfn 8415 . . 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 10629 . . 3 1 ∈ V
12 fnconstg 6542 . . 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 3623 . . . . . . 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 2908 . . . . . 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 7706 . . . . 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 7706 . . . 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 8415 . . 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 6646 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
2625breq2d 5043 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
2726ifbid 4447 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
2827csbeq1d 3832 . . . . . . . . . . . 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 6651 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
30 2fveq3 6651 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
3130imaeq1d 5896 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
3231xpeq1d 5549 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
3330imaeq1d 5896 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
3433xpeq1d 5549 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
3532, 34uneq12d 4091 . . . . . . . . . . . . . 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 7154 . . . . . . . . . . . . 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 3835 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 5127 . . . . . . . . . 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 2809 . . . . . . . . 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 3631 . . . . . . . 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 500 . . . . . . 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 5036 . . . . . . . . . . . 12 ((𝑦 = (𝑁 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4644, 45sylan2 595 . . . . . . . . . . 11 ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4746ancoms 462 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
48 oveq1 7143 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
493nncnd 11644 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
50 npcan1 11057 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5149, 50syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
5248, 51sylan9eqr 2855 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
5347, 52ifbieq2d 4450 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
545nn0ge0d 11949 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑁 − 1))
55 0red 10636 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℝ)
565nn0red 11947 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5755, 56lenltd 10778 . . . . . . . . . . . 12 (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
5854, 57mpbid 235 . . . . . . . . . . 11 (𝜑 → ¬ (𝑁 − 1) < 0)
5958iffalsed 4436 . . . . . . . . . 10 (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6059adantr 484 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6153, 60eqtrd 2833 . . . . . . . 8 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
6261csbeq1d 3832 . . . . . . 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 7144 . . . . . . . . . . . . . . 15 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
6463imaeq2d 5897 . . . . . . . . . . . . . 14 (𝑗 = 𝑁 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
65 xp2nd 7707 . . . . . . . . . . . . . . . . 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 6659 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
68 f1oeq1 6580 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6967, 68elab 3615 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
7066, 69sylib 221 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1ofo 6598 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
72 foima 6571 . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = (1...𝑁))
7574xpeq1d 5549 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76 oveq1 7143 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
7776oveq1d 7151 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
783nnred 11643 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
7978ltp1d 11562 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 < (𝑁 + 1))
803nnzd 12077 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
8180peano2zd 12081 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 + 1) ∈ ℤ)
82 fzn 12921 . . . . . . . . . . . . . . . . . 18 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8381, 80, 82syl2anc 587 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
8479, 83mpbid 235 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
8577, 84sylan9eqr 2855 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
8685imaeq2d 5897 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ∅))
8786xpeq1d 5549 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ∅) × {0}))
88 ima0 5913 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
8988xpeq1i 5546 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = (∅ × {0})
90 0xp 5614 . . . . . . . . . . . . . 14 (∅ × {0}) = ∅
9189, 90eqtri 2821 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = ∅
9287, 91eqtrdi 2849 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
9375, 92uneq12d 4091 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94 un0 4298 . . . . . . . . . . 11 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
9593, 94eqtrdi 2849 . . . . . . . . . 10 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
9695oveq2d 7152 . . . . . . . . 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 3864 . . . . . . . 8 (𝜑𝑁 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1})))
9897adantr 484 . . . . . . 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 2833 . . . . . 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 7171 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
10143, 99, 7, 100fvmptd 6753 . . . . 5 (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1})))
102101fveq1d 6648 . . . 4 (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
103102adantr 484 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
104 inidm 4145 . . . 4 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105 eqidd 2799 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
10611fvconst2 6944 . . . . 5 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107106adantl 485 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
10824, 13, 1, 1, 104, 105, 107ofval 7400 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
109103, 108eqtrd 2833 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
110 elmapi 8414 . . . . . . 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 6829 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
113 elfzonn0 13080 . . . . 5 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
114112, 113syl 17 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
115114nn0cnd 11948 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
116 pncan1 11056 . . 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 7413 1 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3441  ⦋csb 3828   ∪ cun 3879  ∅c0 4243  ifcif 4425  {csn 4525   class class class wbr 5031   ↦ cmpt 5111   × cxp 5518   “ cima 5523   Fn wfn 6320  ⟶wf 6321  –onto→wfo 6323  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136   ∘f cof 7389  1st c1st 7672  2nd c2nd 7673   ↑m cmap 8392  ℂcc 10527  0cc0 10529  1c1 10530   + caddc 10532   < clt 10667   ≤ cle 10668   − cmin 10862  ℕcn 11628  ℕ0cn0 11888  ℤcz 11972  ...cfz 12888  ..^cfzo 13031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-of 7391  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-fzo 13032 This theorem is referenced by:  poimirlem11  35087  poimirlem13  35089
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