Theorem poimirlem8 38737

Description: Lemma for poimir 38762, establishing that away from the opposite vertex the walks in poimirlem9 38738 yield the same vertices. (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)⟶1-1onto(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)
poimirlem9.2	⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
poimirlem9.3	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Assertion

Ref	Expression
poimirlem8	⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))

Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝑈,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑈,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑡,𝑈   𝑆,𝑗,𝑡,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem8

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimirlem9.3	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
2		elrabi 3644	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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}))))} → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
3		poimirlem22.s	. . . . . . . . 9 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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}))))}
4	2, 3	eleq2s 2882	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
5	1, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
6		xp1st 8054	. . . . . . 7 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
8		xp2nd 8055	. . . . . 6 ⊢ ((1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
9	7, 8	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
10		fvex 6918	. . . . . 6 ⊢ (2nd ‘(1st ‘𝑈)) ∈ V
11		f1oeq1 6831	. . . . . 6 ⊢ (𝑓 = (2nd ‘(1st ‘𝑈)) → (𝑓:(1...𝑁)⟶1-1onto(1...𝑁) ↔ (2nd ‘(1st ‘𝑈)):(1...𝑁)⟶1-1onto(1...𝑁)))
12	10, 11	elab 3636	. . . . 5 ⊢ ((2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)} ↔ (2nd ‘(1st ‘𝑈)):(1...𝑁)⟶1-1onto(1...𝑁))
13	9, 12	sylib 221	. . . 4 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)):(1...𝑁)⟶1-1onto(1...𝑁))
14		f1ofn 6844	. . . 4 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)⟶1-1onto(1...𝑁) → (2nd ‘(1st ‘𝑈)) Fn (1...𝑁))
15	13, 14	syl 18	. . 3 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) Fn (1...𝑁))
16		difss 4075	. . 3 ⊢ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ⊆ (1...𝑁)
17		fnssres 6681	. . 3 ⊢ (((2nd ‘(1st ‘𝑈)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ⊆ (1...𝑁)) → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
18	15, 16, 17	sylancl 598	. 2 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
19		poimirlem9.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
20		elrabi 3644	. . . . . . . . 9 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
21	20, 3	eleq2s 2882	. . . . . . . 8 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
22	19, 21	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
23		xp1st 8054	. . . . . . 7 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
24	22, 23	syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
25		xp2nd 8055	. . . . . 6 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
26	24, 25	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
27		fvex 6918	. . . . . 6 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
28		f1oeq1 6831	. . . . . 6 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)⟶1-1onto(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)⟶1-1onto(1...𝑁)))
29	27, 28	elab 3636	. . . . 5 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)⟶1-1onto(1...𝑁))
30	26, 29	sylib 221	. . . 4 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)⟶1-1onto(1...𝑁))
31		f1ofn 6844	. . . 4 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)⟶1-1onto(1...𝑁) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
32	30, 31	syl 18	. . 3 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
33		fnssres 6681	. . 3 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ⊆ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
34	32, 16, 33	sylancl 598	. 2 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
35		poimirlem9.2	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
36		fzp1elp1 13735	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
38		poimir.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
39	38	nncnd 12375	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
40		npcan1 11765	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
41	39, 40	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
42	41	oveq2d 7456	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
43	37, 42	eleqtrd 2866	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ (1...𝑁))
44		fzsplit 13708	. . . . . . . . . 10 ⊢ (((2nd ‘𝑇) + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...((2nd ‘𝑇) + 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
45	43, 44	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = ((1...((2nd ‘𝑇) + 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
46	45	difeq1d 4065	. . . . . . . 8 ⊢ (𝜑 → ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (((1...((2nd ‘𝑇) + 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
47		difundir 4239	. . . . . . . . 9 ⊢ (((1...((2nd ‘𝑇) + 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (((1...((2nd ‘𝑇) + 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
48		elfznn 13711	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ∈ ℕ)
49	35, 48	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℕ)
50	49	nncnd 12375	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℂ)
51		npcan1 11765	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑇) ∈ ℂ → (((2nd ‘𝑇) − 1) + 1) = (2nd ‘𝑇))
52	50, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘𝑇) − 1) + 1) = (2nd ‘𝑇))
53		nnuz 13028	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
54	49, 53	eleqtrdi 2874	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (ℤ≥‘1))
55	52, 54	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2nd ‘𝑇) − 1) + 1) ∈ (ℤ≥‘1))
56	49	nnzd 12743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℤ)
57		peano2zm 12763	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑇) ∈ ℤ → ((2nd ‘𝑇) − 1) ∈ ℤ)
58	56, 57	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘𝑇) − 1) ∈ ℤ)
59		uzid 13004	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘𝑇) − 1) ∈ ℤ → ((2nd ‘𝑇) − 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
60		peano2uz 13052	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘𝑇) − 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)) → (((2nd ‘𝑇) − 1) + 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
61	58, 59, 60	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2nd ‘𝑇) − 1) + 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
62	52, 61	eqeltrrd 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
63		peano2uz 13052	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑇) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)) → ((2nd ‘𝑇) + 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
64	62, 63	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1)))
65		fzsplit2 13707	. . . . . . . . . . . . . 14 ⊢ (((((2nd ‘𝑇) − 1) + 1) ∈ (ℤ≥‘1) ∧ ((2nd ‘𝑇) + 1) ∈ (ℤ≥‘((2nd ‘𝑇) − 1))) → (1...((2nd ‘𝑇) + 1)) = ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) − 1) + 1)...((2nd ‘𝑇) + 1))))
66	55, 64, 65	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...((2nd ‘𝑇) + 1)) = ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) − 1) + 1)...((2nd ‘𝑇) + 1))))
67	52	oveq1d 7455	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((((2nd ‘𝑇) − 1) + 1)...((2nd ‘𝑇) + 1)) = ((2nd ‘𝑇)...((2nd ‘𝑇) + 1)))
68		fzpr 13737	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑇) ∈ ℤ → ((2nd ‘𝑇)...((2nd ‘𝑇) + 1)) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
69	56, 68	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘𝑇)...((2nd ‘𝑇) + 1)) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
70	67, 69	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2nd ‘𝑇) − 1) + 1)...((2nd ‘𝑇) + 1)) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
71	70	uneq2d 4170	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) − 1) + 1)...((2nd ‘𝑇) + 1))) = ((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
72	66, 71	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...((2nd ‘𝑇) + 1)) = ((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
73	72	difeq1d 4065	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...((2nd ‘𝑇) + 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
74	49	nnred 12374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℝ)
75	74	ltm1d 12273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑇) − 1) < (2nd ‘𝑇))
76	58	zred 12827	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘𝑇) − 1) ∈ ℝ)
77	76, 74	ltnled 11481	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2nd ‘𝑇) − 1) < (2nd ‘𝑇) ↔ ¬ (2nd ‘𝑇) ≤ ((2nd ‘𝑇) − 1)))
78	75, 77	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (2nd ‘𝑇) ≤ ((2nd ‘𝑇) − 1))
79		elfzle2 13685	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑇) ∈ (1...((2nd ‘𝑇) − 1)) → (2nd ‘𝑇) ≤ ((2nd ‘𝑇) − 1))
80	78, 79	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (2nd ‘𝑇) ∈ (1...((2nd ‘𝑇) − 1)))
81		difsn 4761	. . . . . . . . . . . . . 14 ⊢ (¬ (2nd ‘𝑇) ∈ (1...((2nd ‘𝑇) − 1)) → ((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇)}) = (1...((2nd ‘𝑇) − 1)))
82	80, 81	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇)}) = (1...((2nd ‘𝑇) − 1)))
83		peano2re 11507	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑇) ∈ ℝ → ((2nd ‘𝑇) + 1) ∈ ℝ)
84	74, 83	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ ℝ)
85	74	ltp1d 12271	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘𝑇) < ((2nd ‘𝑇) + 1))
86	76, 74, 84, 75, 85	lttrd 11495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑇) − 1) < ((2nd ‘𝑇) + 1))
87	76, 84	ltnled 11481	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2nd ‘𝑇) − 1) < ((2nd ‘𝑇) + 1) ↔ ¬ ((2nd ‘𝑇) + 1) ≤ ((2nd ‘𝑇) − 1)))
88	86, 87	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((2nd ‘𝑇) + 1) ≤ ((2nd ‘𝑇) − 1))
89		elfzle2 13685	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑇) + 1) ∈ (1...((2nd ‘𝑇) − 1)) → ((2nd ‘𝑇) + 1) ≤ ((2nd ‘𝑇) − 1))
90	88, 89	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ ((2nd ‘𝑇) + 1) ∈ (1...((2nd ‘𝑇) − 1)))
91		difsn 4761	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑇) + 1) ∈ (1...((2nd ‘𝑇) − 1)) → ((1...((2nd ‘𝑇) − 1)) ∖ {((2nd ‘𝑇) + 1)}) = (1...((2nd ‘𝑇) − 1)))
92	90, 91	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2nd ‘𝑇) − 1)) ∖ {((2nd ‘𝑇) + 1)}) = (1...((2nd ‘𝑇) − 1)))
93	82, 92	ineq12d 4121	. . . . . . . . . . . 12 ⊢ (𝜑 → (((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇)}) ∩ ((1...((2nd ‘𝑇) − 1)) ∖ {((2nd ‘𝑇) + 1)})) = ((1...((2nd ‘𝑇) − 1)) ∩ (1...((2nd ‘𝑇) − 1))))
94		difun2 4440	. . . . . . . . . . . . 13 ⊢ (((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
95		df-pr 4591	. . . . . . . . . . . . . 14 ⊢ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} = ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)})
96	95	difeq2i 4063	. . . . . . . . . . . . 13 ⊢ ((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((1...((2nd ‘𝑇) − 1)) ∖ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)}))
97		difundi 4238	. . . . . . . . . . . . 13 ⊢ ((1...((2nd ‘𝑇) − 1)) ∖ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)})) = (((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇)}) ∩ ((1...((2nd ‘𝑇) − 1)) ∖ {((2nd ‘𝑇) + 1)}))
98	94, 96, 97	3eqtrri 2791	. . . . . . . . . . . 12 ⊢ (((1...((2nd ‘𝑇) − 1)) ∖ {(2nd ‘𝑇)}) ∩ ((1...((2nd ‘𝑇) − 1)) ∖ {((2nd ‘𝑇) + 1)})) = (((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
99		inidm 4126	. . . . . . . . . . . 12 ⊢ ((1...((2nd ‘𝑇) − 1)) ∩ (1...((2nd ‘𝑇) − 1))) = (1...((2nd ‘𝑇) − 1))
100	93, 98, 99	3eqtr3g 2821	. . . . . . . . . . 11 ⊢ (𝜑 → (((1...((2nd ‘𝑇) − 1)) ∪ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (1...((2nd ‘𝑇) − 1)))
101	73, 100	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝜑 → ((1...((2nd ‘𝑇) + 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (1...((2nd ‘𝑇) − 1)))
102		peano2re 11507	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘𝑇) + 1) ∈ ℝ → (((2nd ‘𝑇) + 1) + 1) ∈ ℝ)
103	84, 102	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2nd ‘𝑇) + 1) + 1) ∈ ℝ)
104	84	ltp1d 12271	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) < (((2nd ‘𝑇) + 1) + 1))
105	74, 84, 103, 85, 104	lttrd 11495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘𝑇) < (((2nd ‘𝑇) + 1) + 1))
106	74, 103	ltnled 11481	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘𝑇) < (((2nd ‘𝑇) + 1) + 1) ↔ ¬ (((2nd ‘𝑇) + 1) + 1) ≤ (2nd ‘𝑇)))
107	105, 106	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2nd ‘𝑇) + 1) + 1) ≤ (2nd ‘𝑇))
108		elfzle1 13684	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑇) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁) → (((2nd ‘𝑇) + 1) + 1) ≤ (2nd ‘𝑇))
109	107, 108	nsyl 141	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (2nd ‘𝑇) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))
110		difsn 4761	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑇) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁) → (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇)}) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
111	109, 110	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇)}) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
112	84, 103	ltnled 11481	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘𝑇) + 1) < (((2nd ‘𝑇) + 1) + 1) ↔ ¬ (((2nd ‘𝑇) + 1) + 1) ≤ ((2nd ‘𝑇) + 1)))
113	104, 112	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2nd ‘𝑇) + 1) + 1) ≤ ((2nd ‘𝑇) + 1))
114		elfzle1 13684	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑇) + 1) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁) → (((2nd ‘𝑇) + 1) + 1) ≤ ((2nd ‘𝑇) + 1))
115	113, 114	nsyl 141	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ((2nd ‘𝑇) + 1) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))
116		difsn 4761	. . . . . . . . . . . . 13 ⊢ (¬ ((2nd ‘𝑇) + 1) ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁) → (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2nd ‘𝑇) + 1)}) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
117	115, 116	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2nd ‘𝑇) + 1)}) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
118	111, 117	ineq12d 4121	. . . . . . . . . . 11 ⊢ (𝜑 → ((((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇)}) ∩ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2nd ‘𝑇) + 1)})) = (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
119	95	difeq2i 4063	. . . . . . . . . . . 12 ⊢ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)}))
120		difundi 4238	. . . . . . . . . . . 12 ⊢ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)})) = ((((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇)}) ∩ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2nd ‘𝑇) + 1)}))
121	119, 120	eqtr2i 2787	. . . . . . . . . . 11 ⊢ ((((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇)}) ∩ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2nd ‘𝑇) + 1)})) = (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
122		inidm 4126	. . . . . . . . . . 11 ⊢ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) = ((((2nd ‘𝑇) + 1) + 1)...𝑁)
123	118, 121, 122	3eqtr3g 2821	. . . . . . . . . 10 ⊢ (𝜑 → (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
124	101, 123	uneq12d 4171	. . . . . . . . 9 ⊢ (𝜑 → (((1...((2nd ‘𝑇) + 1)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ (((((2nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
125	47, 124	eqtrid 2810	. . . . . . . 8 ⊢ (𝜑 → (((1...((2nd ‘𝑇) + 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
126	46, 125	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
127	126	eleq2d 2850	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁))))
128		elun 4155	. . . . . 6 ⊢ (𝑘 ∈ ((1...((2nd ‘𝑇) − 1)) ∪ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) ↔ (𝑘 ∈ (1...((2nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
129	127, 128	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ↔ (𝑘 ∈ (1...((2nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))))
130	129	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → (𝑘 ∈ (1...((2nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
131		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
132	131	breq2d 5118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
133	132	ifbid 4509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
134	133	csbeq1d 3855	. . . . . . . . . . . . . . . . . . 19 ⊢ (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}))))
135		2fveq3 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
136		2fveq3 6910	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
137	136	imaeq1d 6062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
138	137	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
139	136	imaeq1d 6062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
140	139	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
141	138, 140	uneq12d 4171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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})))
142	135, 141	oveq12d 7458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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}))))
143	142	csbeq2dv 3858	. . . . . . . . . . . . . . . . . . 19 ⊢ (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}))))
144	134, 143	eqtrd 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (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}))))
145	144	mpteq2dv 5202	. . . . . . . . . . . . . . . . 17 ⊢ (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})))))
146	145	eqeq2d 2774	. . . . . . . . . . . . . . . 16 ⊢ (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}))))))
147	146, 3	elrab2 3652	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(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}))))))
148	147	simprbi 503	. . . . . . . . . . . . . 14 ⊢ (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})))))
149	19, 148	syl 18	. . . . . . . . . . . . 13 ⊢ (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})))))
150		xp1st 8054	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
151	24, 150	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
152		elmapi 8894	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
153	151, 152	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
154		elfzoelz 13817	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155	154	ssriv 3917	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ⊆ ℤ
156		fss 6745	. . . . . . . . . . . . . 14 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
157	153, 155, 156	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
158	38, 149, 157, 30, 35	poimirlem1 38730	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑇))‘𝑛))
159	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → 𝑁 ∈ ℕ)
160		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (2nd ‘𝑡) = (2nd ‘𝑈))
161	160	breq2d 5118	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑈)))
162	161	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)))
163	162	csbeq1d 3855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑈 → ⦋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 ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
164		2fveq3 6910	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑈)))
165		2fveq3 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑈 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑈)))
166	165	imaeq1d 6062	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑈)) “ (1...𝑗)))
167	166	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}))
168	165	imaeq1d 6062	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
169	168	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
170	167, 169	uneq12d 4171	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))
171	164, 170	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → ((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
172	171	csbeq2dv 3858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
173	163, 172	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑈 → ⦋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 ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
174	173	mpteq2dv 5202	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑈 → (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 ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
175	174	eqeq2d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑈 → (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 ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
176	175, 3	elrab2 3652	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
177	176	simprbi 503	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
178	1, 177	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
179	178	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑈)) ∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
180		xp1st 8054	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
181	7, 180	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
182		elmapi 8894	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
183	181, 182	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1st ‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
184		fss 6745	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑈)):(1...𝑁)⟶ℤ)
185	183, 155, 184	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1st ‘(1st ‘𝑈)):(1...𝑁)⟶ℤ)
186	185	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → (1st ‘(1st ‘𝑈)):(1...𝑁)⟶ℤ)
187	13	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → (2nd ‘(1st ‘𝑈)):(1...𝑁)⟶1-1onto(1...𝑁))
188	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
189		xp2nd 8055	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑈) ∈ (0...𝑁))
190	5, 189	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ‘𝑈) ∈ (0...𝑁))
191		eldifsn 4748	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2nd ‘𝑇)}) ↔ ((2nd ‘𝑈) ∈ (0...𝑁) ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)))
192	191	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑈) ∈ (0...𝑁) ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → (2nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2nd ‘𝑇)}))
193	190, 192	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → (2nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2nd ‘𝑇)}))
194	159, 179, 186, 187, 188, 193	poimirlem2 38731	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2nd ‘𝑈) ≠ (2nd ‘𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑇))‘𝑛))
195	194	ex 418	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘𝑈) ≠ (2nd ‘𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑇))‘𝑛)))
196	195	necon1bd 2977	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑇))‘𝑛) → (2nd ‘𝑈) = (2nd ‘𝑇)))
197	158, 196	mpd 16	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑈) = (2nd ‘𝑇))
198	197	oveq1d 7455	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ‘𝑈) − 1) = ((2nd ‘𝑇) − 1))
199	198	oveq2d 7456	. . . . . . . . 9 ⊢ (𝜑 → (1...((2nd ‘𝑈) − 1)) = (1...((2nd ‘𝑇) − 1)))
200	199	eleq2d 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((2nd ‘𝑈) − 1)) ↔ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))))
201	200	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2nd ‘𝑈) − 1)))
202	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑈) − 1))) → 𝑁 ∈ ℕ)
203	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑈) − 1))) → 𝑈 ∈ 𝑆)
204	197, 35	eqeltrd 2864	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝑈) ∈ (1...(𝑁 − 1)))
205	204	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑈) − 1))) → (2nd ‘𝑈) ∈ (1...(𝑁 − 1)))
206		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑈) − 1))) → 𝑘 ∈ (1...((2nd ‘𝑈) − 1)))
207	202, 3, 203, 205, 206	poimirlem6 38735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑈) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2nd ‘(1st ‘𝑈))‘𝑘))
208	201, 207	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2nd ‘(1st ‘𝑈))‘𝑘))
209	38	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → 𝑁 ∈ ℕ)
210	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → 𝑇 ∈ 𝑆)
211	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
212		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2nd ‘𝑇) − 1)))
213	209, 3, 210, 211, 212	poimirlem6 38735	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑘))
214	208, 213	eqtr3d 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2nd ‘𝑇) − 1))) → ((2nd ‘(1st ‘𝑈))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
215	197	oveq1d 7455	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘𝑈) + 1) = ((2nd ‘𝑇) + 1))
216	215	oveq1d 7455	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑈) + 1) + 1) = (((2nd ‘𝑇) + 1) + 1))
217	216	oveq1d 7455	. . . . . . . . 9 ⊢ (𝜑 → ((((2nd ‘𝑈) + 1) + 1)...𝑁) = ((((2nd ‘𝑇) + 1) + 1)...𝑁))
218	217	eleq2d 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)))
219	218	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁))
220	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
221	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑈 ∈ 𝑆)
222	204	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁)) → (2nd ‘𝑈) ∈ (1...(𝑁 − 1)))
223		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁))
224	220, 3, 221, 222, 223	poimirlem7 38736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑈) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑈))‘𝑘))
225	219, 224	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑈))‘𝑘))
226	38	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
227	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑇 ∈ 𝑆)
228	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
229		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))
230	226, 3, 227, 228, 229	poimirlem7 38736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑘))
231	225, 230	eqtr3d 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) → ((2nd ‘(1st ‘𝑈))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
232	214, 231	jaodan 972	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...((2nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))) → ((2nd ‘(1st ‘𝑈))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
233	130, 232	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → ((2nd ‘(1st ‘𝑈))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
234		fvres 6924	. . . 4 ⊢ (𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) → (((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st ‘𝑈))‘𝑘))
235	234	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → (((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st ‘𝑈))‘𝑘))
236		fvres 6924	. . . 4 ⊢ (𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) → (((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
237	236	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → (((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘) = ((2nd ‘(1st ‘𝑇))‘𝑘))
238	233, 235, 237	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → (((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘) = (((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))‘𝑘))
239	18, 34, 238	eqfnfvd 7052	1 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃*wrmo 3368  {crab 3416  ⦋csb 3851   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ifcif 4485  {csn 4588  {cpr 4590   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653   ↾ cres 5657   “ cima 5658   Fn wfn 6550  ⟶wf 6551  ⟶1-1ontowf1o 6554  ‘cfv 6555  ℩crio 7396  (class class class)co 7440   ∘_f cof 7712  1^st c1st 8020  2^nd c2nd 8021   ↑_m cmap 8872  ℂcc 11222  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   < clt 11367   ≤ cle 11368   − cmin 11565  ℕcn 12359  2c2 12421  ℤcz 12717  ℤ_≥cuz 12989  ...cfz 13663  ..^cfzo 13812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813

This theorem is used by:  poimirlem9  38738