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Theorem poimirlem5 35880
Description: Lemma for poimir 35908 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 6819 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 5101 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4495 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
54csbeq1d 3846 . . . . . . . . 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 2fveq3 6824 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
7 2fveq3 6824 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
87imaeq1d 5992 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
98xpeq1d 5643 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
107imaeq1d 5992 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1110xpeq1d 5643 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
129, 11uneq12d 4110 . . . . . . . . . . 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})))
136, 12oveq12d 7347 . . . . . . . . . 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}))))
1413csbeq2dv 3849 . . . . . . . . 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}))))
155, 14eqtrd 2776 . . . . . . . 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}))))
1615mpteq2dv 5191 . . . . . . 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})))))
1716eqeq2d 2747 . . . . . 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}))))))
18 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}))))}
1917, 18elrab2 3637 . . . . 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}))))))
2019simprbi 497 . . . 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})))))
211, 20syl 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})))))
22 breq1 5092 . . . . . . 7 (𝑦 = 0 → (𝑦 < (2nd𝑇) ↔ 0 < (2nd𝑇)))
23 id 22 . . . . . . 7 (𝑦 = 0 → 𝑦 = 0)
2422, 23ifbieq1d 4496 . . . . . 6 (𝑦 = 0 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑇), 0, (𝑦 + 1)))
25 poimirlem5.2 . . . . . . 7 (𝜑 → 0 < (2nd𝑇))
2625iftrued 4480 . . . . . 6 (𝜑 → if(0 < (2nd𝑇), 0, (𝑦 + 1)) = 0)
2724, 26sylan9eqr 2798 . . . . 5 ((𝜑𝑦 = 0) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 0)
2827csbeq1d 3846 . . . 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}))))
29 c0ex 11062 . . . . . . 7 0 ∈ V
30 oveq2 7337 . . . . . . . . . . . . 13 (𝑗 = 0 → (1...𝑗) = (1...0))
31 fz10 13370 . . . . . . . . . . . . 13 (1...0) = ∅
3230, 31eqtrdi 2792 . . . . . . . . . . . 12 (𝑗 = 0 → (1...𝑗) = ∅)
3332imaeq2d 5993 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ ∅))
3433xpeq1d 5643 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ ∅) × {1}))
35 oveq1 7336 . . . . . . . . . . . . . 14 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
36 0p1e1 12188 . . . . . . . . . . . . . 14 (0 + 1) = 1
3735, 36eqtrdi 2792 . . . . . . . . . . . . 13 (𝑗 = 0 → (𝑗 + 1) = 1)
3837oveq1d 7344 . . . . . . . . . . . 12 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
3938imaeq2d 5993 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
4039xpeq1d 5643 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
4134, 40uneq12d 4110 . . . . . . . . 9 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})))
42 ima0 6009 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) “ ∅) = ∅
4342xpeq1i 5640 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = (∅ × {1})
44 0xp 5710 . . . . . . . . . . . 12 (∅ × {1}) = ∅
4543, 44eqtri 2764 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = ∅
4645uneq1i 4105 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
47 uncom 4099 . . . . . . . . . 10 (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
48 un0 4336 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
4946, 47, 483eqtri 2768 . . . . . . . . 9 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5041, 49eqtrdi 2792 . . . . . . . 8 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
5150oveq2d 7345 . . . . . . 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})))
5229, 51csbie 3878 . . . . . 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}))
53 elrabi 3628 . . . . . . . . . . . . . . 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)))
5453, 18eleq2s 2855 . . . . . . . . . . . . . 14 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
551, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
56 xp1st 7923 . . . . . . . . . . . . 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)}))
5755, 56syl 17 . . . . . . . . . . . 12 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
58 xp2nd 7924 . . . . . . . . . . . 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)})
5957, 58syl 17 . . . . . . . . . . 11 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
60 fvex 6832 . . . . . . . . . . . 12 (2nd ‘(1st𝑇)) ∈ V
61 f1oeq1 6749 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6260, 61elab 3619 . . . . . . . . . . 11 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
6359, 62sylib 217 . . . . . . . . . 10 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
64 f1ofo 6768 . . . . . . . . . 10 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
6563, 64syl 17 . . . . . . . . 9 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
66 foima 6738 . . . . . . . . 9 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
6765, 66syl 17 . . . . . . . 8 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
6867xpeq1d 5643 . . . . . . 7 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
6968oveq2d 7345 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘f + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
7052, 69eqtrid 2788 . . . . 5 (𝜑0 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
7170adantr 481 . . . 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})))
7228, 71eqtrd 2776 . . 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})))
73 poimir.0 . . . . 5 (𝜑𝑁 ∈ ℕ)
74 nnm1nn0 12367 . . . . 5 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
7573, 74syl 17 . . . 4 (𝜑 → (𝑁 − 1) ∈ ℕ0)
76 0elfz 13446 . . . 4 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
7775, 76syl 17 . . 3 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
78 ovexd 7364 . . 3 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})) ∈ V)
7921, 72, 77, 78fvmptd 6932 . 2 (𝜑 → (𝐹‘0) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
80 ovexd 7364 . . 3 (𝜑 → (1...𝑁) ∈ V)
81 xp1st 7923 . . . . 5 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
8257, 81syl 17 . . . 4 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
83 elmapfn 8716 . . . 4 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
8482, 83syl 17 . . 3 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
85 fnconstg 6707 . . . 4 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
8629, 85mp1i 13 . . 3 (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
87 eqidd 2737 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
8829fvconst2 7129 . . . 4 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
8988adantl 482 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
90 elmapi 8700 . . . . . . . 8 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9182, 90syl 17 . . . . . . 7 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9291ffvelcdmda 7011 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
93 elfzonn0 13525 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9492, 93syl 17 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9594nn0cnd 12388 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
9695addid1d 11268 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
9780, 84, 86, 84, 87, 89, 96offveq 7611 . 2 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})) = (1st ‘(1st𝑇)))
9879, 97eqtrd 2776 1 (𝜑 → (𝐹‘0) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  {cab 2713  {crab 3403  Vcvv 3441  csb 3842  cun 3895  c0 4268  ifcif 4472  {csn 4572   class class class wbr 5089  cmpt 5172   × cxp 5612  cima 5617   Fn wfn 6468  wf 6469  ontowfo 6471  1-1-ontowf1o 6472  cfv 6473  (class class class)co 7329  f cof 7585  1st c1st 7889  2nd c2nd 7890  m cmap 8678  0cc0 10964  1c1 10965   + caddc 10967   < clt 11102  cmin 11298  cn 12066  0cn0 12326  ...cfz 13332  ..^cfzo 13475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-of 7587  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476
This theorem is referenced by:  poimirlem12  35887  poimirlem14  35889
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