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Theorem poimirlem10 36088
Description: Lemma for poimir 36111 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 7392 . 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 12454 . . . . . 6 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
53, 4syl 17 . . . . 5 (𝜑 → (𝑁 − 1) ∈ ℕ0)
6 nn0fz0 13539 . . . . 5 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
75, 6sylib 217 . . . 4 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
82, 7ffvelcdmd 7036 . . 3 (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9 elmapfn 8803 . . 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 11151 . . 3 1 ∈ V
12 fnconstg 6730 . . 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 3639 . . . . . . 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 2856 . . . . . 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 7953 . . . . 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 7953 . . . 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 8803 . . 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 6842 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
2625breq2d 5117 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
2726ifbid 4509 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
2827csbeq1d 3859 . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
30 2fveq3 6847 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
3130imaeq1d 6012 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
3231xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
3330imaeq1d 6012 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
3433xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
3532, 34uneq12d 4124 . . . . . . . . . . . . . 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 7375 . . . . . . . . . . . . 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 3862 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 5207 . . . . . . . . . 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 2747 . . . . . . . . 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 3648 . . . . . . . 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 5110 . . . . . . . . . . . 12 ((𝑦 = (𝑁 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4644, 45sylan2 593 . . . . . . . . . . 11 ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
4746ancoms 459 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑁 − 1) < 0))
48 oveq1 7364 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
493nncnd 12169 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
50 npcan1 11580 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5149, 50syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
5248, 51sylan9eqr 2798 . . . . . . . . . 10 ((𝜑𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
5347, 52ifbieq2d 4512 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
545nn0ge0d 12476 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑁 − 1))
55 0red 11158 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℝ)
565nn0red 12474 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5755, 56lenltd 11301 . . . . . . . . . . . 12 (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
5854, 57mpbid 231 . . . . . . . . . . 11 (𝜑 → ¬ (𝑁 − 1) < 0)
5958iffalsed 4497 . . . . . . . . . 10 (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6059adantr 481 . . . . . . . . 9 ((𝜑𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
6153, 60eqtrd 2776 . . . . . . . 8 ((𝜑𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
6261csbeq1d 3859 . . . . . . 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 7365 . . . . . . . . . . . . . . 15 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
6463imaeq2d 6013 . . . . . . . . . . . . . 14 (𝑗 = 𝑁 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
65 xp2nd 7954 . . . . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
68 f1oeq1 6772 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6967, 68elab 3630 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
7066, 69sylib 217 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1ofo 6791 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
72 foima 6761 . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = (1...𝑁))
7574xpeq1d 5662 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
7776oveq1d 7372 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
783nnred 12168 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
7978ltp1d 12085 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 < (𝑁 + 1))
803nnzd 12526 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
8180peano2zd 12610 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 + 1) ∈ ℤ)
82 fzn 13457 . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
8577, 84sylan9eqr 2798 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
8685imaeq2d 6013 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑁) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ∅))
8786xpeq1d 5662 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ∅) × {0}))
88 ima0 6029 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) “ ∅) = ∅
8988xpeq1i 5659 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = (∅ × {0})
90 0xp 5730 . . . . . . . . . . . . . 14 (∅ × {0}) = ∅
9189, 90eqtri 2764 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ∅) × {0}) = ∅
9287, 91eqtrdi 2792 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑁) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
9375, 92uneq12d 4124 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94 un0 4350 . . . . . . . . . . 11 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
9593, 94eqtrdi 2792 . . . . . . . . . 10 ((𝜑𝑗 = 𝑁) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
9695oveq2d 7373 . . . . . . . . 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 3893 . . . . . . . 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 2776 . . . . . 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 7392 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
10143, 99, 7, 100fvmptd 6955 . . . . 5 (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1})))
102101fveq1d 6844 . . . 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 4178 . . . 4 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105 eqidd 2737 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
10611fvconst2 7153 . . . . 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 7628 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
109103, 108eqtrd 2776 . 2 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st𝑇))‘𝑛) + 1))
110 elmapi 8787 . . . . . . 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))
112111ffvelcdmda 7035 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
113 elfzonn0 13617 . . . . 5 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
114112, 113syl 17 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
115114nn0cnd 12475 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
116 pncan1 11579 . . 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 7641 1 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  {crab 3407  Vcvv 3445  csb 3855  cun 3908  c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  cima 5636   Fn wfn 6491  wf 6492  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  f cof 7615  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  0cn0 12413  cz 12499  ...cfz 13424  ..^cfzo 13567
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  poimirlem11  36089  poimirlem13  36091
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