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Theorem poimirlem5 37154
Description: Lemma for poimir 37182 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 6891 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 5155 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4547 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
54csbeq1d 3889 . . . . . . . . 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 6896 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
7 2fveq3 6896 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
87imaeq1d 6057 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
98xpeq1d 5701 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
107imaeq1d 6057 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1110xpeq1d 5701 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
129, 11uneq12d 4157 . . . . . . . . . . 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 7433 . . . . . . . . . 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 3892 . . . . . . . . 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 2765 . . . . . . . 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 5245 . . . . . . 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 2736 . . . . . 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 3678 . . . . 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 495 . . . 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 5146 . . . . . . 7 (𝑦 = 0 → (𝑦 < (2nd𝑇) ↔ 0 < (2nd𝑇)))
23 id 22 . . . . . . 7 (𝑦 = 0 → 𝑦 = 0)
2422, 23ifbieq1d 4548 . . . . . 6 (𝑦 = 0 → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑇), 0, (𝑦 + 1)))
25 poimirlem5.2 . . . . . . 7 (𝜑 → 0 < (2nd𝑇))
2625iftrued 4532 . . . . . 6 (𝜑 → if(0 < (2nd𝑇), 0, (𝑦 + 1)) = 0)
2724, 26sylan9eqr 2787 . . . . 5 ((𝜑𝑦 = 0) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 0)
2827csbeq1d 3889 . . . 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 11236 . . . . . . 7 0 ∈ V
30 oveq2 7423 . . . . . . . . . . . . 13 (𝑗 = 0 → (1...𝑗) = (1...0))
31 fz10 13552 . . . . . . . . . . . . 13 (1...0) = ∅
3230, 31eqtrdi 2781 . . . . . . . . . . . 12 (𝑗 = 0 → (1...𝑗) = ∅)
3332imaeq2d 6058 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ ∅))
3433xpeq1d 5701 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ ∅) × {1}))
35 oveq1 7422 . . . . . . . . . . . . . 14 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
36 0p1e1 12362 . . . . . . . . . . . . . 14 (0 + 1) = 1
3735, 36eqtrdi 2781 . . . . . . . . . . . . 13 (𝑗 = 0 → (𝑗 + 1) = 1)
3837oveq1d 7430 . . . . . . . . . . . 12 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
3938imaeq2d 6058 . . . . . . . . . . 11 (𝑗 = 0 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (1...𝑁)))
4039xpeq1d 5701 . . . . . . . . . 10 (𝑗 = 0 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
4134, 40uneq12d 4157 . . . . . . . . 9 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})))
42 ima0 6075 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) “ ∅) = ∅
4342xpeq1i 5698 . . . . . . . . . . . 12 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = (∅ × {1})
44 0xp 5770 . . . . . . . . . . . 12 (∅ × {1}) = ∅
4543, 44eqtri 2753 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) “ ∅) × {1}) = ∅
4645uneq1i 4152 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
47 uncom 4146 . . . . . . . . . 10 (∅ ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
48 un0 4386 . . . . . . . . . 10 ((((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
4946, 47, 483eqtri 2757 . . . . . . . . 9 ((((2nd ‘(1st𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})
5041, 49eqtrdi 2781 . . . . . . . 8 (𝑗 = 0 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}))
5150oveq2d 7431 . . . . . . 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 3921 . . . . . 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 3669 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . 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 8021 . . . . . . . . . . . . 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 8022 . . . . . . . . . . . 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 6904 . . . . . . . . . . . 12 (2nd ‘(1st𝑇)) ∈ V
61 f1oeq1 6821 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
6260, 61elab 3660 . . . . . . . . . . 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 6840 . . . . . . . . . 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 6810 . . . . . . . . 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 5701 . . . . . . 7 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
6968oveq2d 7431 . . . . . 6 (𝜑 → ((1st ‘(1st𝑇)) ∘f + (((2nd ‘(1st𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
7052, 69eqtrid 2777 . . . . 5 (𝜑0 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
7170adantr 479 . . . 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 2765 . . 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 12541 . . . . 5 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
7573, 74syl 17 . . . 4 (𝜑 → (𝑁 − 1) ∈ ℕ0)
76 0elfz 13628 . . . 4 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
7775, 76syl 17 . . 3 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
78 ovexd 7450 . . 3 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})) ∈ V)
7921, 72, 77, 78fvmptd 7006 . 2 (𝜑 → (𝐹‘0) = ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})))
80 ovexd 7450 . . 3 (𝜑 → (1...𝑁) ∈ V)
81 xp1st 8021 . . . . 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 8880 . . . 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 6779 . . . 4 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
8629, 85mp1i 13 . . 3 (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
87 eqidd 2726 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
8829fvconst2 7211 . . . 4 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
8988adantl 480 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
90 elmapi 8864 . . . . . . . 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 7088 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
93 elfzonn0 13707 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9492, 93syl 17 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9594nn0cnd 12562 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
9695addridd 11442 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
9780, 84, 86, 84, 87, 89, 96offveq 7706 . 2 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((1...𝑁) × {0})) = (1st ‘(1st𝑇)))
9879, 97eqtrd 2765 1 (𝜑 → (𝐹‘0) = (1st ‘(1st𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  {crab 3419  Vcvv 3463  csb 3885  cun 3938  c0 4318  ifcif 4524  {csn 4624   class class class wbr 5143  cmpt 5226   × cxp 5670  cima 5675   Fn wfn 6537  wf 6538  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7415  f cof 7679  1st c1st 7987  2nd c2nd 7988  m cmap 8841  0cc0 11136  1c1 11137   + caddc 11139   < clt 11276  cmin 11472  cn 12240  0cn0 12500  ...cfz 13514  ..^cfzo 13657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658
This theorem is referenced by:  poimirlem12  37161  poimirlem14  37163
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