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Theorem poimirlem5 33747
Description: Lemma for poimir 33775 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 6331 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 4795 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4244 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
54csbeq1d 3686 . . . . . . . . 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 6331 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
76fveq2d 6335 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
86fveq2d 6335 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
98imaeq1d 5605 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
109xpeq1d 5277 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
118imaeq1d 5605 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1211xpeq1d 5277 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
1310, 12uneq12d 3916 . . . . . . . . . . 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 6809 . . . . . . . . . 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 4133 . . . . . . . . 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 2803 . . . . . . . 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 4876 . . . . . . 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 2779 . . . . . 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 3515 . . . . 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 484 . . . 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 4786 . . . . . . 7 (𝑦 = 0 → (𝑦 < (2nd𝑇) ↔ 0 < (2nd𝑇)))
24 id 22 . . . . . . 7 (𝑦 = 0 → 𝑦 = 0)
2523, 24ifbieq1d 4245 . . . . . 6 (𝑦 = 0 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑇), 0, (𝑦 + 1)))
26 poimirlem5.2 . . . . . . 7 (𝜑 → 0 < (2nd𝑇))
2726iftrued 4230 . . . . . 6 (𝜑 → if(0 < (2nd𝑇), 0, (𝑦 + 1)) = 0)
2825, 27sylan9eqr 2825 . . . . 5 ((𝜑𝑦 = 0) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 0)
2928csbeq1d 3686 . . . 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 10234 . . . . . . 7 0 ∈ V
31 oveq2 6799 . . . . . . . . . . . . 13 (𝑗 = 0 → (1...𝑗) = (1...0))
32 fz10 12568 . . . . . . . . . . . . 13 (1...0) = ∅
3331, 32syl6eq 2819 . . . . . . . . . . . 12 (𝑗 = 0 → (1...𝑗) = ∅)
3433imaeq2d 5606 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ ∅))
3534xpeq1d 5277 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ ∅) × {1}))
36 oveq1 6798 . . . . . . . . . . . . . 14 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
37 0p1e1 11332 . . . . . . . . . . . . . 14 (0 + 1) = 1
3836, 37syl6eq 2819 . . . . . . . . . . . . 13 (𝑗 = 0 → (𝑗 + 1) = 1)
3938oveq1d 6806 . . . . . . . . . . . 12 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
4039imaeq2d 5606 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
4140xpeq1d 5277 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
4235, 41uneq12d 3916 . . . . . . . . 9 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})))
43 ima0 5621 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) “ ∅) = ∅
4443xpeq1i 5274 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = (∅ × {1})
45 0xp 5338 . . . . . . . . . . . 12 (∅ × {1}) = ∅
4644, 45eqtri 2791 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = ∅
4746uneq1i 3911 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
48 uncom 3905 . . . . . . . . . 10 (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
49 un0 4108 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5047, 48, 493eqtri 2795 . . . . . . . . 9 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5142, 50syl6eq 2819 . . . . . . . 8 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
5251oveq2d 6807 . . . . . . 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 3705 . . . . . 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 3507 . . . . . . . . . . . . . . 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 2866 . . . . . . . . . . . . . 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 7345 . . . . . . . . . . . . 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 7346 . . . . . . . . . . . 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 6341 . . . . . . . . . . . 12 (2nd ‘(1st𝑇)) ∈ V
62 f1oeq1 6267 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6361, 62elab 3498 . . . . . . . . . . 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 6284 . . . . . . . . . 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 6260 . . . . . . . . 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 5277 . . . . . . 7 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
7069oveq2d 6807 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
7153, 70syl5eq 2815 . . . . 5 (𝜑0 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
7271adantr 473 . . . 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 2803 . . 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 11534 . . . . 5 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
7674, 75syl 17 . . . 4 (𝜑 → (𝑁 − 1) ∈ ℕ0)
77 0elfz 12643 . . . 4 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
7876, 77syl 17 . . 3 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
79 ovexd 6823 . . 3 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V)
8022, 73, 78, 79fvmptd 6429 . 2 (𝜑 → (𝐹‘0) = ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
81 ovexd 6823 . . 3 (𝜑 → (1...𝑁) ∈ V)
82 xp1st 7345 . . . . 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 8030 . . . 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 6232 . . . 4 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
8730, 86mp1i 13 . . 3 (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
88 eqidd 2770 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
8930fvconst2 6611 . . . 4 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
9089adantl 474 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
91 elmapi 8029 . . . . . . . 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 6501 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
94 elfzonn0 12720 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9593, 94syl 17 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9695nn0cnd 11553 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
9796addid1d 10436 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
9881, 85, 87, 85, 88, 90, 97offveq 7063 . 2 (𝜑 → ((1st ‘(1st𝑇)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st𝑇)))
9980, 98eqtrd 2803 1 (𝜑 → (𝐹‘0) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wcel 2143  {cab 2755  {crab 3063  Vcvv 3348  csb 3679  cun 3718  c0 4060  ifcif 4222  {csn 4313   class class class wbr 4783  cmpt 4860   × cxp 5246  cima 5251   Fn wfn 6025  wf 6026  ontowfo 6028  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6791  𝑓 cof 7040  1st c1st 7311  2nd c2nd 7312  𝑚 cmap 8007  0cc0 10136  1c1 10137   + caddc 10139   < clt 10274  cmin 10466  cn 11220  0cn0 11492  ...cfz 12532  ..^cfzo 12672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  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-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  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 11888  df-fz 12533  df-fzo 12673
This theorem is referenced by:  poimirlem12  33754  poimirlem14  33756
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