Theorem poimirlem5 36083

Description: Lemma for poimir 36111 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 6842	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 5117	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4509	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3859	. . . . . . . . 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		2fveq3 6847	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
7		2fveq3 6847	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
8	7	imaeq1d 6012	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
9	8	xpeq1d 5662	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
10	7	imaeq1d 6012	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
11	10	xpeq1d 5662	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
12	9, 11	uneq12d 4124	. . . . . . . . . . 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})))
13	6, 12	oveq12d 7375	. . . . . . . . . 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}))))
14	13	csbeq2dv 3862	. . . . . . . . 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}))))
15	5, 14	eqtrd 2776	. . . . . . . 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}))))
16	15	mpteq2dv 5207	. . . . . . 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})))))
17	16	eqeq2d 2747	. . . . . 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}))))))
18		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}))))}
19	17, 18	elrab2 3648	. . . . 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}))))))
20	19	simprbi 497	. . . 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})))))
21	1, 20	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})))))
22		breq1 5108	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < (2nd ‘𝑇) ↔ 0 < (2nd ‘𝑇)))
23		id 22	. . . . . . 7 ⊢ (𝑦 = 0 → 𝑦 = 0)
24	22, 23	ifbieq1d 4510	. . . . . 6 ⊢ (𝑦 = 0 → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)))
25		poimirlem5.2	. . . . . . 7 ⊢ (𝜑 → 0 < (2nd ‘𝑇))
26	25	iftrued 4494	. . . . . 6 ⊢ (𝜑 → if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)) = 0)
27	24, 26	sylan9eqr 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 0) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 0)
28	27	csbeq1d 3859	. . . 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}))))
29		c0ex 11149	. . . . . . 7 ⊢ 0 ∈ V
30		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
31		fz10 13462	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
32	30, 31	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
33	32	imaeq2d 6013	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
34	33	xpeq1d 5662	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}))
35		oveq1 7364	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
36		0p1e1 12275	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
38	37	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
39	38	imaeq2d 6013	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
40	39	xpeq1d 5662	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
41	34, 40	uneq12d 4124	. . . . . . . . 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})))
42		ima0 6029	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
43	42	xpeq1i 5659	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = (∅ × {1})
44		0xp 5730	. . . . . . . . . . . 12 ⊢ (∅ × {1}) = ∅
45	43, 44	eqtri 2764	. . . . . . . . . . 11 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = ∅
46	45	uneq1i 4119	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
47		uncom 4113	. . . . . . . . . 10 ⊢ (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
48		un0 4350	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
49	46, 47, 48	3eqtri 2768	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
50	41, 49	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑗 = 0 → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
51	50	oveq2d 7373	. . . . . . 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})))
52	29, 51	csbie 3891	. . . . . 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}))
53		elrabi 3639	. . . . . . . . . . . . . . 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)))
54	53, 18	eleq2s 2856	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
55	1, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
56		xp1st 7953	. . . . . . . . . . . . 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)}))
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
58		xp2nd 7954	. . . . . . . . . . . 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)})
59	57, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
60		fvex 6855	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
61		f1oeq1 6772	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
62	60, 61	elab 3630	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
63	59, 62	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
64		f1ofo 6791	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
65	63, 64	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
66		foima 6761	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
67	65, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
68	67	xpeq1d 5662	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
69	68	oveq2d 7373	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})))
70	52, 69	eqtrid 2788	. . . . 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})))
71	70	adantr 481	. . . 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})))
72	28, 71	eqtrd 2776	. . 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})))
73		poimir.0	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
74		nnm1nn0 12454	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
75	73, 74	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
76		0elfz 13538	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
77	75, 76	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
78		ovexd 7392	. . 3 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})) ∈ V)
79	21, 72, 77, 78	fvmptd 6955	. 2 ⊢ (𝜑 → (𝐹‘0) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})))
80		ovexd 7392	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ V)
81		xp1st 7953	. . . . 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)))
82	57, 81	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
83		elmapfn 8803	. . . 4 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
84	82, 83	syl 17	. . 3 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
85		fnconstg 6730	. . . 4 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
86	29, 85	mp1i 13	. . 3 ⊢ (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
87		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
88	29	fvconst2 7153	. . . 4 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
89	88	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
90		elmapi 8787	. . . . . . . 8 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
91	82, 90	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
92	91	ffvelcdmda 7035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
93		elfzonn0 13617	. . . . . 6 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
94	92, 93	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
95	94	nn0cnd 12475	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
96	95	addid1d 11355	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + 0) = ((1st ‘(1st ‘𝑇))‘𝑛))
97	80, 84, 86, 84, 87, 89, 96	offveq 7641	. 2 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})) = (1st ‘(1st ‘𝑇)))
98	79, 97	eqtrd 2776	1 ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  {crab 3407  Vcvv 3445  ⦋csb 3855   ∪ cun 3908  ∅c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631   “ cima 5636   Fn wfn 6491  ⟶wf 6492  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615  1^st c1st 7919  2^nd c2nd 7920   ↑_m cmap 8765  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189   − cmin 11385  ℕcn 12153  ℕ₀cn0 12413  ...cfz 13424  ..^cfzo 13567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568

This theorem is referenced by:  poimirlem12  36090  poimirlem14  36092