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Theorem poimirlem5 33544
Description: Lemma for poimir 33572 to establish that, for the simplices defined by a walk along the edges of an 𝑁-cube, if the starting vertex is not opposite a given face, it is the earliest vertex of the face 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}))))}
poimirlem9.1 (𝜑𝑇𝑆)
poimirlem5.2 (𝜑 → 0 < (2nd𝑇))
Assertion
Ref Expression
poimirlem5 (𝜑 → (𝐹‘0) = (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 poimirlem5
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 poimirlem9.1 . . . 4 (𝜑𝑇𝑆)
2 fveq2 6229 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 4697 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4141 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
54csbeq1d 3573 . . . . . . . . 9 (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}))))
6 fveq2 6229 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
76fveq2d 6233 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
86fveq2d 6233 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
98imaeq1d 5500 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
109xpeq1d 5172 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
118imaeq1d 5500 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1211xpeq1d 5172 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
1310, 12uneq12d 3801 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
147, 13oveq12d 6708 . . . . . . . . . 10 (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}))))
1514csbeq2dv 4025 . . . . . . . . 9 (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}))))
165, 15eqtrd 2685 . . . . . . . 8 (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}))))
1716mpteq2dv 4778 . . . . . . 7 (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})))))
1817eqeq2d 2661 . . . . . 6 (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}))))))
19 poimirlem22.s . . . . . 6 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}))))}
2018, 19elrab2 3399 . . . . 5 (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}))))))
2120simprbi 479 . . . 4 (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})))))
221, 21syl 17 . . 3 (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})))))
23 breq1 4688 . . . . . . 7 (𝑦 = 0 → (𝑦 < (2nd𝑇) ↔ 0 < (2nd𝑇)))
24 id 22 . . . . . . 7 (𝑦 = 0 → 𝑦 = 0)
2523, 24ifbieq1d 4142 . . . . . 6 (𝑦 = 0 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑇), 0, (𝑦 + 1)))
26 poimirlem5.2 . . . . . . 7 (𝜑 → 0 < (2nd𝑇))
2726iftrued 4127 . . . . . 6 (𝜑 → if(0 < (2nd𝑇), 0, (𝑦 + 1)) = 0)
2825, 27sylan9eqr 2707 . . . . 5 ((𝜑𝑦 = 0) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 0)
2928csbeq1d 3573 . . . 4 ((𝜑𝑦 = 0) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 0 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
30 c0ex 10072 . . . . . . 7 0 ∈ V
31 oveq2 6698 . . . . . . . . . . . . 13 (𝑗 = 0 → (1...𝑗) = (1...0))
32 fz10 12400 . . . . . . . . . . . . 13 (1...0) = ∅
3331, 32syl6eq 2701 . . . . . . . . . . . 12 (𝑗 = 0 → (1...𝑗) = ∅)
3433imaeq2d 5501 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ ∅))
3534xpeq1d 5172 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ ∅) × {1}))
36 oveq1 6697 . . . . . . . . . . . . . 14 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
37 0p1e1 11170 . . . . . . . . . . . . . 14 (0 + 1) = 1
3836, 37syl6eq 2701 . . . . . . . . . . . . 13 (𝑗 = 0 → (𝑗 + 1) = 1)
3938oveq1d 6705 . . . . . . . . . . . 12 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
4039imaeq2d 5501 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
4140xpeq1d 5172 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
4235, 41uneq12d 3801 . . . . . . . . 9 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})))
43 ima0 5516 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) “ ∅) = ∅
4443xpeq1i 5169 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = (∅ × {1})
45 0xp 5233 . . . . . . . . . . . 12 (∅ × {1}) = ∅
4644, 45eqtri 2673 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = ∅
4746uneq1i 3796 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
48 uncom 3790 . . . . . . . . . 10 (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
49 un0 4000 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5047, 48, 493eqtri 2677 . . . . . . . . 9 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5142, 50syl6eq 2701 . . . . . . . 8 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
5251oveq2d 6706 . . . . . . 7 (𝑗 = 0 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})))
5330, 52csbie 3592 . . . . . 6 0 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
54 elrabi 3391 . . . . . . . . . . . . . . 15 (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)))
5554, 19eleq2s 2748 . . . . . . . . . . . . . 14 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
561, 55syl 17 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
57 xp1st 7242 . . . . . . . . . . . . 13 (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)}))
5856, 57syl 17 . . . . . . . . . . . 12 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
59 xp2nd 7243 . . . . . . . . . . . 12 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
6058, 59syl 17 . . . . . . . . . . 11 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
61 fvex 6239 . . . . . . . . . . . 12 (2nd ‘(1st𝑇)) ∈ V
62 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6361, 62elab 3382 . . . . . . . . . . 11 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
6460, 63sylib 208 . . . . . . . . . 10 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
65 f1ofo 6182 . . . . . . . . . 10 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
6664, 65syl 17 . . . . . . . . 9 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
67 foima 6158 . . . . . . . . 9 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
6866, 67syl 17 . . . . . . . 8 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
6968xpeq1d 5172 . . . . . . 7 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
7069oveq2d 6706 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
7153, 70syl5eq 2697 . . . . 5 (𝜑0 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
7271adantr 480 . . . 4 ((𝜑𝑦 = 0) → 0 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
7329, 72eqtrd 2685 . . 3 ((𝜑𝑦 = 0) → 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) × {0})))
74 poimir.0 . . . . 5 (𝜑𝑁 ∈ ℕ)
75 nnm1nn0 11372 . . . . 5 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
7674, 75syl 17 . . . 4 (𝜑 → (𝑁 − 1) ∈ ℕ0)
77 0elfz 12475 . . . 4 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
7876, 77syl 17 . . 3 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
79 ovexd 6720 . . 3 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V)
8022, 73, 78, 79fvmptd 6327 . 2 (𝜑 → (𝐹‘0) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
81 ovexd 6720 . . 3 (𝜑 → (1...𝑁) ∈ V)
82 xp1st 7242 . . . . 5 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
8358, 82syl 17 . . . 4 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
84 elmapfn 7922 . . . 4 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
8583, 84syl 17 . . 3 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
86 fnconstg 6131 . . . 4 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
8730, 86mp1i 13 . . 3 (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
88 eqidd 2652 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
8930fvconst2 6510 . . . 4 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
9089adantl 481 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
91 elmapi 7921 . . . . . . . 8 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9283, 91syl 17 . . . . . . 7 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9392ffvelrnda 6399 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
94 elfzonn0 12552 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9593, 94syl 17 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9695nn0cnd 11391 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
9796addid1d 10274 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
9881, 85, 87, 85, 88, 90, 97offveq 6960 . 2 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st𝑇)))
9980, 98eqtrd 2685 1 (𝜑 → (𝐹‘0) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  {crab 2945  Vcvv 3231  csb 3566  cun 3605  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  cima 5146   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cmin 10304  cn 11058  0cn0 11330  ...cfz 12364  ..^cfzo 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505
This theorem is referenced by:  poimirlem12  33551  poimirlem14  33553
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