Theorem poimirlem10 35714

Description: Lemma for poimir 35737 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 7290	. 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 12204	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
6		nn0fz0 13283	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
7	5, 6	sylib 217	. . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
8	2, 7	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) ∈ ((0...𝐾) ↑m (1...𝑁)))
9		elmapfn 8611	. . 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 10902	. . 3 ⊢ 1 ∈ V
12		fnconstg 6646	. . 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 3611	. . . . . . 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 2857	. . . . . 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 7836	. . . . 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 7836	. . . 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 8611	. . 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 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
26	25	breq2d 5082	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
27	26	ifbid 4479	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
28	27	csbeq1d 3832	. . . . . . . . . . . 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 6761	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
30		2fveq3 6761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
31	30	imaeq1d 5957	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
32	31	xpeq1d 5609	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
33	30	imaeq1d 5957	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
34	33	xpeq1d 5609	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
35	32, 34	uneq12d 4094	. . . . . . . . . . . . . 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 7273	. . . . . . . . . . . . 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 3835	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 5172	. . . . . . . . . 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 2749	. . . . . . . . 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 3620	. . . . . . . 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 5075	. . . . . . . . . . . 12 ⊢ ((𝑦 = (𝑁 − 1) ∧ (2nd ‘𝑇) = 0) → (𝑦 < (2nd ‘𝑇) ↔ (𝑁 − 1) < 0))
46	44, 45	sylan2 592	. . . . . . . . . . 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 7262	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
49	3	nncnd 11919	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
50		npcan1 11330	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
52	48, 51	sylan9eqr 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → (𝑦 + 1) = 𝑁)
53	47, 52	ifbieq2d 4482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 0, 𝑦, 𝑁))
54	5	nn0ge0d 12226	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁 − 1))
55		0red 10909	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
56	5	nn0red 12224	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
57	55, 56	lenltd 11051	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0))
58	54, 57	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑁 − 1) < 0)
59	58	iffalsed 4467	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑁)
62	61	csbeq1d 3832	. . . . . . 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 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
64	63	imaeq2d 5958	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑁 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
65		xp2nd 7837	. . . . . . . . . . . . . . . . 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 6769	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
68		f1oeq1 6688	. . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
71		f1ofo 6707	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
72		foima 6677	. . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = (1...𝑁))
75	74	xpeq1d 5609	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = ((1...𝑁) × {1}))
76		oveq1 7262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
77	76	oveq1d 7270	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
78	3	nnred 11918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	ltp1d 11835	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
80	3	nnzd 12354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
81	80	peano2zd 12358	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
82		fzn 13201	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
83	81, 80, 82	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
84	79, 83	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
85	77, 84	sylan9eqr 2801	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((𝑗 + 1)...𝑁) = ∅)
86	85	imaeq2d 5958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ∅))
87	86	xpeq1d 5609	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}))
88		ima0 5974	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
89	88	xpeq1i 5606	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = (∅ × {0})
90		0xp 5675	. . . . . . . . . . . . . 14 ⊢ (∅ × {0}) = ∅
91	89, 90	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) “ ∅) × {0}) = ∅
92	87, 91	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = ∅)
93	75, 92	uneq12d 4094	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
94		un0 4321	. . . . . . . . . . 11 ⊢ (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
95	93, 94	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 = 𝑁) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
96	95	oveq2d 7271	. . . . . . . . 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 3866	. . . . . . . 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 2778	. . . . . 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 7290	. . . . . 6 ⊢ (𝜑 → ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})) ∈ V)
101	43, 99, 7, 100	fvmptd 6864	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑁 − 1)) = ((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1})))
102	101	fveq1d 6758	. . . 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 4149	. . . 4 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
105		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
106	11	fvconst2 7061	. . . . 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 7522	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇)) ∘f + ((1...𝑁) × {1}))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
109	103, 108	eqtrd 2778	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑁 − 1))‘𝑛) = (((1st ‘(1st ‘𝑇))‘𝑛) + 1))
110		elmapi 8595	. . . . . . 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	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
113		elfzonn0 13360	. . . . 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 12225	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
116		pncan1 11329	. . 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 7535	1 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422  ⦋csb 3828   ∪ cun 3881  ∅c0 4253  ifcif 4456  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941   − cmin 11135  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ...cfz 13168  ..^cfzo 13311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312

This theorem is referenced by:  poimirlem11  35715  poimirlem13  35717