Theorem poimirlem5 33414

Description: Lemma for poimir 33442 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 6191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 4665	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4108	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3540	. . . . . . . . 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 6191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇))
7	6	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
8	6	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
9	8	imaeq1d 5465	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
10	9	xpeq1d 5138	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
11	8	imaeq1d 5465	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
12	11	xpeq1d 5138	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
13	10, 12	uneq12d 3768	. . . . . . . . . . 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 6668	. . . . . . . . . 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 3992	. . . . . . . . 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 2656	. . . . . . . 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 4745	. . . . . . 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 2632	. . . . . 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 3366	. . . . 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 480	. . . 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 4656	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < (2nd ‘𝑇) ↔ 0 < (2nd ‘𝑇)))
24		id 22	. . . . . . 7 ⊢ (𝑦 = 0 → 𝑦 = 0)
25	23, 24	ifbieq1d 4109	. . . . . 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 4094	. . . . . 6 ⊢ (𝜑 → if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)) = 0)
28	25, 27	sylan9eqr 2678	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 0) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 0)
29	28	csbeq1d 3540	. . . 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 10034	. . . . . . 7 ⊢ 0 ∈ V
31		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
32		fz10 12362	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
33	31, 32	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
34	33	imaeq2d 5466	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
35	34	xpeq1d 5138	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}))
36		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
37		0p1e1 11132	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
38	36, 37	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
39	38	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
40	39	imaeq2d 5466	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
41	40	xpeq1d 5138	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
42	35, 41	uneq12d 3768	. . . . . . . . 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 5481	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
44	43	xpeq1i 5135	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = (∅ × {1})
45		0xp 5199	. . . . . . . . . . . 12 ⊢ (∅ × {1}) = ∅
46	44, 45	eqtri 2644	. . . . . . . . . . 11 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = ∅
47	46	uneq1i 3763	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
48		uncom 3757	. . . . . . . . . 10 ⊢ (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
49		un0 3967	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
50	47, 48, 49	3eqtri 2648	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
51	42, 50	syl6eq 2672	. . . . . . . 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 6666	. . . . . . 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 3559	. . . . . 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 3359	. . . . . . . . . . . . . . 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 2719	. . . . . . . . . . . . . 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 7198	. . . . . . . . . . . . 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 7199	. . . . . . . . . . . 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 6201	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
62		f1oeq1 6127	. . . . . . . . . . . 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 3350	. . . . . . . . . . 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 208	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
65		f1ofo 6144	. . . . . . . . . 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 6120	. . . . . . . . 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 5138	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
70	69	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
71	53, 70	syl5eq 2668	. . . . 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 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})))
73	29, 72	eqtrd 2656	. . 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 11334	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
76	74, 75	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
77		0elfz 12436	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
78	76, 77	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
79		ovexd 6680	. . 3 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) ∈ V)
80	22, 73, 78, 79	fvmptd 6288	. 2 ⊢ (𝜑 → (𝐹‘0) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})))
81		ovexd 6680	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ V)
82		xp1st 7198	. . . . 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)))
83	58, 82	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
84		elmapfn 7880	. . . 4 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
85	83, 84	syl 17	. . 3 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
86		fnconstg 6093	. . . 4 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
87	30, 86	mp1i 13	. . 3 ⊢ (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
88		eqidd 2623	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
89	30	fvconst2 6469	. . . 4 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
90	89	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
91		elmapi 7879	. . . . . . . 8 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
92	83, 91	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
93	92	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
94		elfzonn0 12512	. . . . . 6 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
95	93, 94	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
96	95	nn0cnd 11353	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
97	96	addid1d 10236	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + 0) = ((1st ‘(1st ‘𝑇))‘𝑛))
98	81, 85, 87, 85, 88, 90, 97	offveq 6918	. 2 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st ‘𝑇)))
99	80, 98	eqtrd 2656	1 ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916  Vcvv 3200  ⦋csb 3533   ∪ cun 3572  ∅c0 3915  ifcif 4086  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ...cfz 12326  ..^cfzo 12465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466

This theorem is referenced by:  poimirlem12  33421  poimirlem14  33423