Theorem poimirlem10 37676

Description: Lemma for poimir 37699 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 7387	. 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 12428	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
6		nn0fz0 13531	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
7	5, 6	sylib 218	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
8	2, 7	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9		elmapfn 8795	. . 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 11114	. . 3 ⊢ 1 ∈ V
12		fnconstg 6717	. . 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 3638	. . . . . . 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 2849	. . . . . 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 7959	. . . . 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 7959	. . . 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 8795	. . 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 6828	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
26	25	breq2d 5105	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
27	26	ifbid 4498	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
28	27	csbeq1d 3849	. . . . . . . . . . . 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 6833	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
30		2fveq3 6833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
31	30	imaeq1d 6013	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
32	31	xpeq1d 5648	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
33	30	imaeq1d 6013	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
34	33	xpeq1d 5648	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
35	32, 34	uneq12d 4118	. . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . 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 3852	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 5187	. . . . . . . . . 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 2742	. . . . . . . . 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 3645	. . . . . . . 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 496	. . . . . . 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 5098	. . . . . . . . . . . 12 ⊢ ((𝑦 = (𝑁 − 1) ∧ (2nd ‘𝑇) = 0) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
46	44, 45	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑦 = (𝑁 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
47	46	ancoms 458	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
48		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
49	3	nncnd 12147	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
50		npcan1 11548	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
52	48, 51	sylan9eqr 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
53	47, 52	ifbieq2d 4501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
54	5	nn0ge0d 12451	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁 − 1))
55		0red 11121	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
56	5	nn0red 12449	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
57	55, 56	lenltd 11265	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
58	54, 57	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑁 − 1) < 0)
59	58	iffalsed 4485	. . . . . . . . . 10 ⊢ (𝜑 → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
60	59	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if((𝑁 − 1) < 0, 𝑦, 𝑁) = 𝑁)
61	53, 60	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
62	61	csbeq1d 3849	. . . . . . 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 7360	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
64	63	imaeq2d 6014	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑁 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
65		xp2nd 7960	. . . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
68		f1oeq1 6757	. . . . . . . . . . . . . . . . 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 3630	. . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71		f1ofo 6776	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
72		foima 6746	. . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = (1...𝑁))
75	74	xpeq1d 5648	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76		oveq1 7359	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
77	76	oveq1d 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
78	3	nnred 12146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	ltp1d 12058	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
80	3	nnzd 12501	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
81	80	peano2zd 12586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
82		fzn 13446	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
83	81, 80, 82	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
84	79, 83	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
85	77, 84	sylan9eqr 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
86	85	imaeq2d 6014	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
87	86	xpeq1d 5648	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}))
88		ima0 6031	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
89	88	xpeq1i 5645	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = (∅ × {0})
90		0xp 5718	. . . . . . . . . . . . . 14 ⊢ (∅ × {0}) = ∅
91	89, 90	eqtri 2754	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = ∅
92	87, 91	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
93	75, 92	uneq12d 4118	. . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
96	95	oveq2d 7368	. . . . . . . . 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 3881	. . . . . . . 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 480	. . . . . . 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 2766	. . . . . 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 7387	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
101	43, 99, 7, 100	fvmptd 6942	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
102	101	fveq1d 6830	. . . 4 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
103	102	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛))
104		inidm 4176	. . . 4 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
106	11	fvconst2 7144	. . . . 5 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {1})‘𝑛) = 1)
107	106	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {1})‘𝑛) = 1)
108	24, 13, 1, 1, 104, 105, 107	ofval 7627	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
109	103, 108	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
110		elmapi 8779	. . . . . . 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 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
113		elfzonn0 13613	. . . . 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 12450	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
116		pncan1 11547	. . 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 7642	1 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  {crab 3395  Vcvv 3436  ⦋csb 3845   ∪ cun 3895  ∅c0 4282  ifcif 4474  {csn 4575   class class class wbr 5093   ↦ cmpt 5174   × cxp 5617   “ cima 5622   Fn wfn 6482  ⟶wf 6483  –onto→wfo 6485  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7352   ∘_f cof 7614  1^st c1st 7925  2^nd c2nd 7926   ↑_m cmap 8756  ℂcc 11010  0cc0 11012  1c1 11013   + caddc 11015   < clt 11152   ≤ cle 11153   − cmin 11350  ℕcn 12131  ℕ₀cn0 12387  ℤcz 12474  ...cfz 13413  ..^cfzo 13560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-n0 12388  df-z 12475  df-uz 12739  df-fz 13414  df-fzo 13561

This theorem is referenced by:  poimirlem11  37677  poimirlem13  37679