Theorem poimirlem5 33859

Description: Lemma for poimir 33887 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 6379	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 4823	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4267	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3700	. . . . . . . . 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 6384	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
7		2fveq3 6384	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
8	7	imaeq1d 5649	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
9	8	xpeq1d 5308	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
10	7	imaeq1d 5649	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
11	10	xpeq1d 5308	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
12	9, 11	uneq12d 3932	. . . . . . . . . . 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 6864	. . . . . . . . . 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 4155	. . . . . . . . 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 2799	. . . . . . . 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 4906	. . . . . . 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 2775	. . . . . 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 3525	. . . . 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 490	. . . 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 4814	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < (2nd ‘𝑇) ↔ 0 < (2nd ‘𝑇)))
23		id 22	. . . . . . 7 ⊢ (𝑦 = 0 → 𝑦 = 0)
24	22, 23	ifbieq1d 4268	. . . . . 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 4253	. . . . . 6 ⊢ (𝜑 → if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)) = 0)
27	24, 26	sylan9eqr 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 0) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 0)
28	27	csbeq1d 3700	. . . 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 10291	. . . . . . 7 ⊢ 0 ∈ V
30		oveq2 6854	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
31		fz10 12574	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
32	30, 31	syl6eq 2815	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
33	32	imaeq2d 5650	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
34	33	xpeq1d 5308	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}))
35		oveq1 6853	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
36		0p1e1 11405	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
37	35, 36	syl6eq 2815	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
38	37	oveq1d 6861	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
39	38	imaeq2d 5650	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
40	39	xpeq1d 5308	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
41	34, 40	uneq12d 3932	. . . . . . . . 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 5665	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
43	42	xpeq1i 5305	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = (∅ × {1})
44		0xp 5371	. . . . . . . . . . . 12 ⊢ (∅ × {1}) = ∅
45	43, 44	eqtri 2787	. . . . . . . . . . 11 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = ∅
46	45	uneq1i 3927	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
47		uncom 3921	. . . . . . . . . 10 ⊢ (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
48		un0 4131	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
49	46, 47, 48	3eqtri 2791	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
50	41, 49	syl6eq 2815	. . . . . . . 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 6862	. . . . . . 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 3719	. . . . . 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 3516	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . 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 7403	. . . . . . . . . . . 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 6392	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
61		f1oeq1 6314	. . . . . . . . . . . 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 3507	. . . . . . . . . . 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 209	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
64		f1ofo 6331	. . . . . . . . . 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 6305	. . . . . . . . 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 5308	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
69	68	oveq2d 6862	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
70	52, 69	syl5eq 2811	. . . . 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 472	. . . 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 2799	. . 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 11585	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
75	73, 74	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
76		0elfz 12649	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
77	75, 76	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
78		ovexd 6880	. . 3 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V)
79	21, 72, 77, 78	fvmptd 6481	. 2 ⊢ (𝜑 → (𝐹‘0) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
80		ovexd 6880	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ V)
81		xp1st 7402	. . . . 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 8087	. . . 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 6277	. . . 4 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
86	29, 85	mp1i 13	. . 3 ⊢ (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
87		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
88	29	fvconst2 6666	. . . 4 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
89	88	adantl 473	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
90		elmapi 8086	. . . . . . . 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	ffvelrnda 6553	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
93		elfzonn0 12726	. . . . . 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 11604	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
96	95	addid1d 10494	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + 0) = ((1st ‘(1st ‘𝑇))‘𝑛))
97	80, 84, 86, 84, 87, 89, 96	offveq 7120	. 2 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st ‘𝑇)))
98	79, 97	eqtrd 2799	1 ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  {cab 2751  {crab 3059  Vcvv 3350  ⦋csb 3693   ∪ cun 3732  ∅c0 4081  ifcif 4245  {csn 4336   class class class wbr 4811   ↦ cmpt 4890   × cxp 5277   “ cima 5282   Fn wfn 6065  ⟶wf 6066  –onto→wfo 6068  –1-1-onto→wf1o 6069  ‘cfv 6070  (class class class)co 6846   ∘_𝑓 cof 7097  1^st c1st 7368  2^nd c2nd 7369   ↑_𝑚 cmap 8064  0cc0 10193  1c1 10194   + caddc 10196   < clt 10332   − cmin 10524  ℕcn 11278  ℕ₀cn0 11542  ...cfz 12538  ..^cfzo 12678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-n0 11543  df-z 11629  df-uz 11892  df-fz 12539  df-fzo 12679

This theorem is referenced by:  poimirlem12  33866  poimirlem14  33868