Theorem poimirlem5 36481

Description: Lemma for poimir 36509 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 6888	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 5159	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4550	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3896	. . . . . . . . 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 6893	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
7		2fveq3 6893	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
8	7	imaeq1d 6056	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
9	8	xpeq1d 5704	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
10	7	imaeq1d 6056	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
11	10	xpeq1d 5704	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
12	9, 11	uneq12d 4163	. . . . . . . . . . 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 7423	. . . . . . . . . 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 3899	. . . . . . . . 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 2772	. . . . . . . 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 5249	. . . . . . 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 2743	. . . . . 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 3685	. . . . 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 5150	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < (2nd ‘𝑇) ↔ 0 < (2nd ‘𝑇)))
23		id 22	. . . . . . 7 ⊢ (𝑦 = 0 → 𝑦 = 0)
24	22, 23	ifbieq1d 4551	. . . . . 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 4535	. . . . . 6 ⊢ (𝜑 → if(0 < (2nd ‘𝑇), 0, (𝑦 + 1)) = 0)
27	24, 26	sylan9eqr 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 0) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 0)
28	27	csbeq1d 3896	. . . 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 11204	. . . . . . 7 ⊢ 0 ∈ V
30		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
31		fz10 13518	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
32	30, 31	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
33	32	imaeq2d 6057	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
34	33	xpeq1d 5704	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}))
35		oveq1 7412	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
36		0p1e1 12330	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
38	37	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
39	38	imaeq2d 6057	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
40	39	xpeq1d 5704	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
41	34, 40	uneq12d 4163	. . . . . . . . 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 6073	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
43	42	xpeq1i 5701	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = (∅ × {1})
44		0xp 5772	. . . . . . . . . . . 12 ⊢ (∅ × {1}) = ∅
45	43, 44	eqtri 2760	. . . . . . . . . . 11 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) = ∅
46	45	uneq1i 4158	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}))
47		uncom 4152	. . . . . . . . . 10 ⊢ (∅ ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅)
48		un0 4389	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
49	46, 47, 48	3eqtri 2764	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ∅) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})
50	41, 49	eqtrdi 2788	. . . . . . . 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 7421	. . . . . . 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 3928	. . . . . 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 3676	. . . . . . . . . . . . . . 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 2851	. . . . . . . . . . . . . 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 8003	. . . . . . . . . . . . 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 8004	. . . . . . . . . . . 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 6901	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
61		f1oeq1 6818	. . . . . . . . . . . 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 3667	. . . . . . . . . . 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 6837	. . . . . . . . . 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 6807	. . . . . . . . 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 5704	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
69	68	oveq2d 7421	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) × {0})) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})))
70	52, 69	eqtrid 2784	. . . . 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 2772	. . 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 12509	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
75	73, 74	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
76		0elfz 13594	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
77	75, 76	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
78		ovexd 7440	. . 3 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})) ∈ V)
79	21, 72, 77, 78	fvmptd 7002	. 2 ⊢ (𝜑 → (𝐹‘0) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})))
80		ovexd 7440	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ V)
81		xp1st 8003	. . . . 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 8855	. . . 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 6776	. . . 4 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
86	29, 85	mp1i 13	. . 3 ⊢ (𝜑 → ((1...𝑁) × {0}) Fn (1...𝑁))
87		eqidd 2733	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
88	29	fvconst2 7201	. . . 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 8839	. . . . . . . 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 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
93		elfzonn0 13673	. . . . . 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 12530	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
96	95	addridd 11410	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + 0) = ((1st ‘(1st ‘𝑇))‘𝑛))
97	80, 84, 86, 84, 87, 89, 96	offveq 7690	. 2 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {0})) = (1st ‘(1st ‘𝑇)))
98	79, 97	eqtrd 2772	1 ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  {crab 3432  Vcvv 3474  ⦋csb 3892   ∪ cun 3945  ∅c0 4321  ifcif 4527  {csn 4627   class class class wbr 5147   ↦ cmpt 5230   × cxp 5673   “ cima 5678   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405   ∘_f cof 7664  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8816  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   − cmin 11440  ℕcn 12208  ℕ₀cn0 12468  ...cfz 13480  ..^cfzo 13623

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624

This theorem is referenced by:  poimirlem12  36488  poimirlem14  36490