Theorem poimirlem10 38549

Description: Lemma for poimir 38572 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 7455	. 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 12647	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
5	3, 4	syl 18	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
6		nn0fz0 13759	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
7	5, 6	sylib 221	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
8	2, 7	ffvelcdmd 7085	. . 3 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9		elmapfn 8887	. . 3 ⊢ ((𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)) → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
10	8, 9	syl 18	. 2 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) Fn (1...𝑁))
11		1ex 11303	. . 3 ⊢ 1 ∈ V
12		fnconstg 6770	. . 3 ⊢ (1 ∈ V → ((1...𝑁) × {1}) Fn (1...𝑁))
13	11, 12	mp1i 14	. 2 ⊢ (𝜑 → ((1...𝑁) × {1}) Fn (1...𝑁))
14		poimirlem12.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
15		elrabi 3640	. . . . . . 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 2878	. . . . . 6 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
18	14, 17	syl 18	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
19		xp1st 8033	. . . . 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 18	. . . 4 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21		xp1st 8033	. . . 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 18	. . 3 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
23		elmapfn 8887	. . 3 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
24	22, 23	syl 18	. 2 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
25		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
26	25	breq2d 5114	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
27	26	ifbid 4505	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
28	27	csbeq1d 3850	. . . . . . . . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
30		2fveq3 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
31	30	imaeq1d 6050	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
32	31	xpeq1d 5679	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
33	30	imaeq1d 6050	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
34	33	xpeq1d 5679	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
35	32, 34	uneq12d 4168	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . 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 3853	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 5198	. . . . . . . . . 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 2771	. . . . . . . . 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 3648	. . . . . . . 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 503	. . . . . . 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 18	. . . . . 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 5107	. . . . . . . . . . . 12 ⊢ ((𝑦 = (𝑁 − 1) ∧ (2nd ‘𝑇) = 0) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
46	44, 45	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
47	46	ancoms 464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
48		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
49	3	nncnd 12351	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
50		npcan1 11741	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
51	49, 50	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
52	48, 51	sylan9eqr 2817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
53	47, 52	ifbieq2d 4508	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
54	5	nn0ge0d 12670	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁 − 1))
55		0red 11311	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
56	5	nn0red 12668	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
57	55, 56	lenltd 11456	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
58	54, 57	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑁 − 1) < 0)
59	58	iffalsed 4492	. . . . . . . . . 10 ⊢ (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
60	59	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
61	53, 60	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
62	61	csbeq1d 3850	. . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
64	63	imaeq2d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑁 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
65		xp2nd 8034	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
67		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
68		f1oeq1 6812	. . . . . . . . . . . . . . . . 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 3632	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71		f1ofo 6832	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
72		foima 6801	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
73	70, 71, 72	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
74	64, 73	sylan9eqr 2817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = (1...𝑁))
75	74	xpeq1d 5679	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76		oveq1 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
77	76	oveq1d 7435	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
78	3	nnred 12350	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	ltp1d 12247	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
80	3	nnzd 12719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
81	80	peano2zd 12806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
82		fzn 13673	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
83	81, 80, 82	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
84	79, 83	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
85	77, 84	sylan9eqr 2817	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
86	85	imaeq2d 6051	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
87	86	xpeq1d 5679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}))
88		ima0 6074	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
89	88	xpeq1i 5676	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = (∅ × {0})
90		0xp 5749	. . . . . . . . . . . . . 14 ⊢ (∅ × {0}) = ∅
91	89, 90	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = ∅
92	87, 91	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
93	75, 92	uneq12d 4168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94		un0 4343	. . . . . . . . . . 11 ⊢ (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
95	93, 94	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
96	95	oveq2d 7436	. . . . . . . . 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 3882	. . . . . . . 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 486	. . . . . . 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 2795	. . . . . 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 7455	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
101	43, 99, 7, 100	fvmptd 7001	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
102	101	fveq1d 6887	. . . 4 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
103	102	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
104		inidm 4123	. . . 4 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
106	11	fvconst2 7210	. . . . 5 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107	106	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
108	24, 13, 1, 1, 104, 105, 107	ofval 7704	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
109	103, 108	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
110		elmapi 8869	. . . . . . 7 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
111	22, 110	syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
112	111	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
113		elfzonn0 13842	. . . . 5 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
114	112, 113	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
115	114	nn0cnd 12669	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
116		pncan1 11740	. . 3 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ → ((((1st ‘(1st ‘𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st ‘𝑇))‘𝑛))
117	115, 116	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st ‘𝑇))‘𝑛) + 1) − 1) = ((1st ‘(1st ‘𝑇))‘𝑛))
118	1, 10, 13, 24, 109, 107, 117	offveq 7719	1 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412  Vcvv 3450  ⦋csb 3846   ∪ cun 4149  ∅c0 4278  ifcif 4481  {csn 4583   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648   “ cima 5653   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8847  ℂcc 11198  0cc0 11200  1c1 11201   + caddc 11203   < clt 11343   ≤ cle 11344   − cmin 11541  ℕcn 12335  ℕ₀cn0 12606  ℤcz 12693  ...cfz 13639  ..^cfzo 13788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789

This theorem is used by:  poimirlem11  38550  poimirlem13  38552