Theorem poimirlem10 38010

Description: Lemma for poimir 38033 establishing the cube that yields the simplex that yields a face if the opposite vertex was first 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}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
poimirlem12.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem11.3	⊢ (𝜑 → (2nd ‘𝑇) = 0)

Assertion

Ref	Expression
poimirlem10	⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (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 poimirlem10

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7394	. 2 ⊢ (𝜑 → (1...𝑁) ∈ V)
2		poimirlem22.1	. . . 4 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
3		poimir.0	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		nnm1nn0 12473	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
6		nn0fz0 13574	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
7	5, 6	sylib 220	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
8	2, 7	ffvelcdmd 7029	. . 3 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9		elmapfn 8806	. . 3 ⊢ ((𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)) → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
11		1ex 11136	. . 3 ⊢ 1 ∈ V
12		fnconstg 6718	. . 3 ⊢ (1 ∈ V → ((1...𝑁) × {1}) Fn (1...𝑁))
13	11, 12	mp1i 13	. 2 ⊢ (𝜑 → ((1...𝑁) × {1}) Fn (1...𝑁))
14		poimirlem12.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
15		elrabi 3626	. . . . . . 7 ⊢ (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)))
16		poimirlem22.s	. . . . . . 7 ⊢ 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}))))}
17	15, 16	eleq2s 2859	. . . . . 6 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
18	14, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
19		xp1st 7965	. . . . 5 ⊢ (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)}))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21		xp1st 7965	. . . 4 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
23		elmapfn 8806	. . 3 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
24	22, 23	syl 17	. 2 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
25		fveq2 6830	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
26	25	breq2d 5086	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
27	26	ifbid 4480	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
28	27	csbeq1d 3836	. . . . . . . . . . . 12 ⊢ (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}))))
29		2fveq3 6835	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
30		2fveq3 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
31	30	imaeq1d 6017	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
32	31	xpeq1d 5649	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
33	30	imaeq1d 6017	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
34	33	xpeq1d 5649	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
35	32, 34	uneq12d 4101	. . . . . . . . . . . . . 14 ⊢ (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})))
36	29, 35	oveq12d 7377	. . . . . . . . . . . . 13 ⊢ (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}))))
37	36	csbeq2dv 3839	. . . . . . . . . . . 12 ⊢ (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}))))
38	28, 37	eqtrd 2776	. . . . . . . . . . 11 ⊢ (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}))))
39	38	mpteq2dv 5168	. . . . . . . . . 10 ⊢ (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})))))
40	39	eqeq2d 2752	. . . . . . . . 9 ⊢ (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}))))))
41	40, 16	elrab2 3633	. . . . . . . 8 ⊢ (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}))))))
42	41	simprbi 499	. . . . . . 7 ⊢ (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})))))
43	14, 42	syl 17	. . . . . 6 ⊢ (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})))))
44		poimirlem11.3	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘𝑇) = 0)
45		breq12 5079	. . . . . . . . . . . 12 ⊢ ((𝑦 = (𝑁 − 1) ∧ (2nd ‘𝑇) = 0) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
46	44, 45	sylan2 600	. . . . . . . . . . 11 ⊢ ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
47	46	ancoms 460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
48		oveq1 7366	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
49	3	nncnd 12185	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
50		npcan1 11571	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
52	48, 51	sylan9eqr 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
53	47, 52	ifbieq2d 4483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
54	5	nn0ge0d 12496	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁 − 1))
55		0red 11143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
56	5	nn0red 12494	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
57	55, 56	lenltd 11288	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
58	54, 57	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑁 − 1) < 0)
59	58	iffalsed 4467	. . . . . . . . . 10 ⊢ (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
60	59	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
61	53, 60	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
62	61	csbeq1d 3836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = (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}))) = ⦋𝑁 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
63		oveq2 7367	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
64	63	imaeq2d 6018	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑁 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
65		xp2nd 7966	. . . . . . . . . . . . . . . . 17 ⊢ ((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)})
66	20, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
67		fvex 6843	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
68		f1oeq1 6758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
69	67, 68	elab 3618	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
70	66, 69	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71		f1ofo 6777	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
72		foima 6747	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
73	70, 71, 72	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
74	64, 73	sylan9eqr 2798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = (1...𝑁))
75	74	xpeq1d 5649	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76		oveq1 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
77	76	oveq1d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
78	3	nnred 12184	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	ltp1d 12081	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
80	3	nnzd 12545	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
81	80	peano2zd 12631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
82		fzn 13489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
83	81, 80, 82	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
84	79, 83	mpbid 234	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
85	77, 84	sylan9eqr 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
86	85	imaeq2d 6018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
87	86	xpeq1d 5649	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}))
88		ima0 6035	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
89	88	xpeq1i 5646	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = (∅ × {0})
90		0xp 5719	. . . . . . . . . . . . . 14 ⊢ (∅ × {0}) = ∅
91	89, 90	eqtri 2764	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = ∅
92	87, 91	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
93	75, 92	uneq12d 4101	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94		un0 4324	. . . . . . . . . . 11 ⊢ (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
95	93, 94	eqtrdi 2792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
96	95	oveq2d 7375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
97	3, 96	csbied 3868	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑁 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
98	97	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → ⦋𝑁 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
99	62, 98	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (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}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
100		ovexd 7394	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
101	43, 99, 7, 100	fvmptd 6946	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
102	101	fveq1d 6832	. . . 4 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
103	102	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
104		inidm 4157	. . . 4 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105		eqidd 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
106	11	fvconst2 7151	. . . . 5 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107	106	adantl 483	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
108	24, 13, 1, 1, 104, 105, 107	ofval 7634	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
109	103, 108	eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
110		elmapi 8790	. . . . . . 7 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
111	22, 110	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
112	111	ffvelcdmda 7028	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
113		elfzonn0 13657	. . . . 5 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
114	112, 113	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
115	114	nn0cnd 12495	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
116		pncan1 11570	. . 3 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ → ((((1st ‘(1st ‘𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st ‘𝑇))‘𝑛))
117	115, 116	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st ‘𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st ‘𝑇))‘𝑛))
118	1, 10, 13, 24, 109, 107, 117	offveq 7649	1 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  {crab 3393  Vcvv 3433  ⦋csb 3832   ∪ cun 3882  ∅c0 4263  ifcif 4456  {csn 4557   class class class wbr 5074   ↦ cmpt 5155   × cxp 5618   “ cima 5623   Fn wfn 6483  ⟶wf 6484  –onto→wfo 6486  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359   ∘_f cof 7621  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8767  ℂcc 11032  0cc0 11034  1c1 11035   + caddc 11037   < clt 11175   ≤ cle 11176   − cmin 11373  ℕcn 12169  ℕ₀cn0 12432  ℤcz 12519  ...cfz 13456  ..^cfzo 13603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604

This theorem is referenced by:  poimirlem11  38011  poimirlem13  38013