Theorem poimirlem5 32584

Description: Lemma for poimir 32612 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	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem9.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem5.2	⊢ (𝜑 → 0 < (2nd ‘𝑇))

Assertion

Ref	Expression
poimirlem5	⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

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.

Step	Hyp	Ref	Expression
1		poimirlem9.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 4595	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4058	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3506	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
6		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇))
7	6	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
8	6	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
9	8	imaeq1d 5384	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
10	9	xpeq1d 5062	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
11	8	imaeq1d 5384	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
12	11	xpeq1d 5062	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
13	10, 12	uneq12d 3730	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
14	7, 13	oveq12d 6567	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
15	14	csbeq2dv 3944	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
16	5, 15	eqtrd 2644	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
17	16	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
18	17	eqeq2d 2620	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
19		poimirlem22.s	. . . . . 6 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
20	18, 19	elrab2 3333	. . . . 5 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
21	20	simprbi 479	. . . 4 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
22	1, 21	syl 17	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
23		breq1 4586	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < (2nd ‘𝑇) ↔ 0 < (2nd ‘𝑇)))
24		id 22	. . . . . . 7 ⊢ (𝑦 = 0 → 𝑦 = 0)
25	23, 24	ifbieq1d 4059	. . . . . 6 ⊢ (𝑦 = 0 → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)))
26		poimirlem5.2	. . . . . . 7 ⊢ (𝜑 → 0 < (2nd ‘𝑇))
27	26	iftrued 4044	. . . . . 6 ⊢ (𝜑 → if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)) = 0)
28	25, 27	sylan9eqr 2666	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 0) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 0)
29	28	csbeq1d 3506	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 0) → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋0 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
30		c0ex 9913	. . . . . . 7 ⊢ 0 ∈ V
31		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
32		fz10 12233	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
33	31, 32	syl6eq 2660	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
34	33	imaeq2d 5385	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
35	34	xpeq1d 5062	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}))
36		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
37		0p1e1 11009	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
38	36, 37	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
39	38	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
40	39	imaeq2d 5385	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
41	40	xpeq1d 5062	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
42	35, 41	uneq12d 3730	. . . . . . . . 9 ⊢ (𝑗 = 0 → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})))
43		ima0 5400	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
44	43	xpeq1i 5059	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = (∅ × {1})
45		0xp 5122	. . . . . . . . . . . 12 ⊢ (∅ × {1}) = ∅
46	44, 45	eqtri 2632	. . . . . . . . . . 11 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = ∅
47	46	uneq1i 3725	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
48		uncom 3719	. . . . . . . . . 10 ⊢ (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
49		un0 3919	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
50	47, 48, 49	3eqtri 2636	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
51	42, 50	syl6eq 2660	. . . . . . . 8 ⊢ (𝑗 = 0 → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
52	51	oveq2d 6565	. . . . . . 7 ⊢ (𝑗 = 0 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})))
53	30, 52	csbie 3525	. . . . . 6 ⊢ ⦋0 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
54		elrabi 3328	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
55	54, 19	eleq2s 2706	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
56	1, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
57		xp1st 7089	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
58	56, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
59		xp2nd 7090	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
61		fvex 6113	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
62		f1oeq1 6040	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
63	61, 62	elab 3319	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
64	60, 63	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
65		f1ofo 6057	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
66	64, 65	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
67		foima 6033	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
68	66, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
69	68	xpeq1d 5062	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
70	69	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
71	53, 70	syl5eq 2656	. . . . 5 ⊢ (𝜑 → ⦋0 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
72	71	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 0) → ⦋0 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
73	29, 72	eqtrd 2644	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 0) → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
74		poimir.0	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
75		nnm1nn0 11211	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
76	74, 75	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
77		0elfz 12305	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
78	76, 77	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
79		ovex 6577	. . . 4 ⊢ ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V
80	79	a1i 11	. . 3 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V)
81	22, 73, 78, 80	fvmptd 6197	. 2 ⊢ (𝜑 → (𝐹‘0) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
82		ovex 6577	. . . 4 ⊢ (1...𝑁) ∈ V
83	82	a1i 11	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ V)
84		xp1st 7089	. . . . 5 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
85	58, 84	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
86		elmapfn 7766	. . . 4 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
87	85, 86	syl 17	. . 3 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
88		fnconstg 6006	. . . 4 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
89	30, 88	mp1i 13	. . 3 ⊢ (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
90		eqidd 2611	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
91	30	fvconst2 6374	. . . 4 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
92	91	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
93		elmapi 7765	. . . . . . . 8 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
94	85, 93	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
95	94	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
96		elfzonn0 12380	. . . . . 6 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
97	95, 96	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
98	97	nn0cnd 11230	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
99	98	addid1d 10115	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + 0) = ((1st ‘(1st ‘𝑇))‘𝑛))
100	83, 87, 89, 87, 90, 92, 99	offveq 6816	. 2 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st ‘𝑇)))
101	81, 100	eqtrd 2644	1 ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  ⦋csb 3499   ∪ cun 3538  ∅c0 3874  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ...cfz 12197  ..^cfzo 12334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335

This theorem is referenced by:  poimirlem12  32591  poimirlem14  32593