poimirlem8 - Mathbox for Brendan Leahy

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Theorem poimirlem8 36986

Description: Lemma for poimir 37011, establishing that away from the opposite vertex the walks in poimirlem9 36987 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)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem9.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem9.2	⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
poimirlem9.3	⊢ (𝜑 → 𝑈 ∈ 𝑆)

**Assertion**
Ref	Expression
poimirlem8	⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘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 3669	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3		poimirlem22.s	. . . . . . . . 9 ⊢ 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 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4	2, 3	eleq2s 2843	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6		xp1st 8000	. . . . . . 7 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8		xp2nd 8001	. . . . . 6 ⊢ ((1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10		fvex 6894	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
11		f1oeq1 6811	. . . . . 6 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
12	10, 11	elab 3660	. . . . 5 ⊢ ((2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
13	9, 12	sylib 217	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14		f1ofn 6824	. . . 4 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁))
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁))
16		difss 4123	. . 3 ⊢ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)
17		fnssres 6663	. . 3 ⊢ (((2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
18	15, 16, 17	sylancl 585	. 2 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
19		poimirlem9.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
20		elrabi 3669	. . . . . . . . 9 ⊢ (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 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
21	20, 3	eleq2s 2843	. . . . . . . 8 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23		xp1st 8000	. . . . . . 7 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25		xp2nd 8001	. . . . . 6 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
26	24, 25	syl 17	. . . . 5 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
27		fvex 6894	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
28		f1oeq1 6811	. . . . . 6 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
29	27, 28	elab 3660	. . . . 5 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
30	26, 29	sylib 217	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31		f1ofn 6824	. . . 4 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
32	30, 31	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
33		fnssres 6663	. . 3 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
34	32, 16, 33	sylancl 585	. 2 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
35		poimirlem9.2	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
36		fzp1elp1 13551	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
38		poimir.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
39	38	nncnd 12225	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
40		npcan1 11636	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
42	41	oveq2d 7417	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
43	37, 42	eleqtrd 2827	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
44		fzsplit 13524	. . . . . . . . . 10 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = ((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
46	45	difeq1d 4113	. . . . . . . 8 ⊢ (𝜑 → ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
47		difundir 4272	. . . . . . . . 9 ⊢ (((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
48		elfznn 13527	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
49	35, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
50	49	nncnd 12225	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℂ)
51		npcan1 11636	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) ∈ ℂ → (((2^nd ‘𝑇) − 1) + 1) = (2^nd ‘𝑇))
52	50, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) = (2^nd ‘𝑇))
53		nnuz 12862	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
54	49, 53	eleqtrdi 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘1))
55	52, 54	eqeltrd 2825	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘1))
56	49	nnzd 12582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℤ)
57		peano2zm 12602	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ ℤ)
58	56, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℤ)
59		uzid 12834	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) − 1) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
60		peano2uz 12882	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
61	58, 59, 60	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
62	52, 61	eqeltrrd 2826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
63		peano2uz 12882	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) → ((2^nd ‘𝑇) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
64	62, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
65		fzsplit2 13523	. . . . . . . . . . . . . 14 ⊢ (((((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘1) ∧ ((2^nd ‘𝑇) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1))) → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))))
66	55, 64, 65	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))))
67	52	oveq1d 7416	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)))
68		fzpr 13553	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
69	56, 68	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
70	67, 69	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
71	70	uneq2d 4155	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))) = ((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
72	66, 71	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
73	72	difeq1d 4113	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
74	49	nnred 12224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
75	74	ltm1d 12143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇))
76	58	zred 12663	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℝ)
77	76, 74	ltnled 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇) ↔ ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1)))
78	75, 77	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
79		elfzle2 13502	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)) → (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
80	78, 79	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)))
81		difsn 4793	. . . . . . . . . . . . . 14 ⊢ (¬ (2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)) → ((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) = (1...((2^nd ‘𝑇) − 1)))
82	80, 81	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) = (1...((2^nd ‘𝑇) − 1)))
83		peano2re 11384	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℝ → ((2^nd ‘𝑇) + 1) ∈ ℝ)
84	74, 83	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℝ)
85	74	ltp1d 12141	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
86	76, 74, 84, 75, 85	lttrd 11372	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1))
87	76, 84	ltnled 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1)))
88	86, 87	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
89		elfzle2 13502	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)) → ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
90	88, 89	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)))
91		difsn 4793	. . . . . . . . . . . . . 14 ⊢ (¬ ((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)) → ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
92	90, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
93	82, 92	ineq12d 4205	. . . . . . . . . . . 12 ⊢ (𝜑 → (((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) ∩ ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)})) = ((1...((2^nd ‘𝑇) − 1)) ∩ (1...((2^nd ‘𝑇) − 1))))
94		difun2 4472	. . . . . . . . . . . . 13 ⊢ (((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
95		df-pr 4623	. . . . . . . . . . . . . 14 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
96	95	difeq2i 4111	. . . . . . . . . . . . 13 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...((2^nd ‘𝑇) − 1)) ∖ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)}))
97		difundi 4271	. . . . . . . . . . . . 13 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∖ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})) = (((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) ∩ ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)}))
98	94, 96, 97	3eqtrri 2757	. . . . . . . . . . . 12 ⊢ (((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) ∩ ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)})) = (((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
99		inidm 4210	. . . . . . . . . . . 12 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∩ (1...((2^nd ‘𝑇) − 1))) = (1...((2^nd ‘𝑇) − 1))
100	93, 98, 99	3eqtr3g 2787	. . . . . . . . . . 11 ⊢ (𝜑 → (((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
101	73, 100	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
102		peano2re 11384	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) + 1) ∈ ℝ → (((2^nd ‘𝑇) + 1) + 1) ∈ ℝ)
103	84, 102	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ∈ ℝ)
104	84	ltp1d 12141	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1))
105	74, 84, 103, 85, 104	lttrd 11372	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1))
106	74, 103	ltnled 11358	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇)))
107	105, 106	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
108		elfzle1 13501	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
109	107, 108	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
110		difsn 4793	. . . . . . . . . . . . 13 ⊢ (¬ (2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
111	109, 110	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
112	84, 103	ltnled 11358	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
113	104, 112	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
114		elfzle1 13501	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
115	113, 114	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
116		difsn 4793	. . . . . . . . . . . . 13 ⊢ (¬ ((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
117	115, 116	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
118	111, 117	ineq12d 4205	. . . . . . . . . . 11 ⊢ (𝜑 → ((((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) ∩ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)})) = (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
119	95	difeq2i 4111	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)}))
120		difundi 4271	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})) = ((((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) ∩ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)}))
121	119, 120	eqtr2i 2753	. . . . . . . . . . 11 ⊢ ((((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) ∩ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)})) = (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
122		inidm 4210	. . . . . . . . . . 11 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁)
123	118, 121, 122	3eqtr3g 2787	. . . . . . . . . 10 ⊢ (𝜑 → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
124	101, 123	uneq12d 4156	. . . . . . . . 9 ⊢ (𝜑 → (((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
125	47, 124	eqtrid 2776	. . . . . . . 8 ⊢ (𝜑 → (((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
126	46, 125	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
127	126	eleq2d 2811	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ↔ 𝑘 ∈ ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))))
128		elun 4140	. . . . . 6 ⊢ (𝑘 ∈ ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) ↔ (𝑘 ∈ (1...((2^nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
129	127, 128	bitrdi 287	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ↔ (𝑘 ∈ (1...((2^nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))))
130	129	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (𝑘 ∈ (1...((2^nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
131		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
132	131	breq2d 5150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
133	132	ifbid 4543	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
134	133	csbeq1d 3889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
135		2fveq3 6886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
136		2fveq3 6886	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
137	136	imaeq1d 6048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
138	137	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
139	136	imaeq1d 6048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
140	139	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
141	138, 140	uneq12d 4156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
142	135, 141	oveq12d 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
143	142	csbeq2dv 3892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
144	134, 143	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
145	144	mpteq2dv 5240	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
146	145	eqeq2d 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
147	146, 3	elrab2 3678	. . . . . . . . . . . . . . 15 ⊢ (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 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
148	147	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
149	19, 148	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
150		xp1st 8000	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
151	24, 150	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
152		elmapi 8839	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
153	151, 152	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
154		elfzoelz 13629	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155	154	ssriv 3978	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ⊆ ℤ
156		fss 6724	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
157	153, 155, 156	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
158	38, 149, 157, 30, 35	poimirlem1 36979	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
159	38	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → 𝑁 ∈ ℕ)
160		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (2^nd ‘𝑡) = (2^nd ‘𝑈))
161	160	breq2d 5150	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑈)))
162	161	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)))
163	162	csbeq1d 3889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
164		2fveq3 6886	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑈)))
165		2fveq3 6886	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑈 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑈)))
166	165	imaeq1d 6048	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)))
167	166	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}))
168	165	imaeq1d 6048	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
169	168	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
170	167, 169	uneq12d 4156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))
171	164, 170	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
172	171	csbeq2dv 3892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
173	163, 172	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
174	173	mpteq2dv 5240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
175	174	eqeq2d 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
176	175, 3	elrab2 3678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
177	176	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
178	1, 177	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
179	178	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
180		xp1st 8000	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
181	7, 180	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
182		elmapi 8839	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
183	181, 182	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
184		fss 6724	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
185	183, 155, 184	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
186	185	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
187	13	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
188	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
189		xp2nd 8001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2^nd ‘𝑈) ∈ (0...𝑁))
190	5, 189	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (0...𝑁))
191		eldifsn 4782	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}) ↔ ((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)))
192	191	biimpri 227	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
193	190, 192	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
194	159, 179, 186, 187, 188, 193	poimirlem2 36980	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
195	194	ex 412	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘𝑈) ≠ (2^nd ‘𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛)))
196	195	necon1bd 2950	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛) → (2^nd ‘𝑈) = (2^nd ‘𝑇)))
197	158, 196	mpd 15	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘𝑈) = (2^nd ‘𝑇))
198	197	oveq1d 7416	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑈) − 1) = ((2^nd ‘𝑇) − 1))
199	198	oveq2d 7417	. . . . . . . . 9 ⊢ (𝜑 → (1...((2^nd ‘𝑈) − 1)) = (1...((2^nd ‘𝑇) − 1)))
200	199	eleq2d 2811	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((2^nd ‘𝑈) − 1)) ↔ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))))
201	200	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
202	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑁 ∈ ℕ)
203	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑈 ∈ 𝑆)
204	197, 35	eqeltrd 2825	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
205	204	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
206		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
207	202, 3, 203, 205, 206	poimirlem6 36984	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
208	201, 207	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
209	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑁 ∈ ℕ)
210	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑇 ∈ 𝑆)
211	35	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
212		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑇) − 1)))
213	209, 3, 210, 211, 212	poimirlem6 36984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 1))‘𝑛) ≠ ((𝐹‘𝑘)‘𝑛)) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
214	208, 213	eqtr3d 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
215	197	oveq1d 7416	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑈) + 1) = ((2^nd ‘𝑇) + 1))
216	215	oveq1d 7416	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘𝑈) + 1) + 1) = (((2^nd ‘𝑇) + 1) + 1))
217	216	oveq1d 7416	. . . . . . . . 9 ⊢ (𝜑 → ((((2^nd ‘𝑈) + 1) + 1)...𝑁) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
218	217	eleq2d 2811	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
219	218	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
220	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
221	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑈 ∈ 𝑆)
222	204	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
223		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
224	220, 3, 221, 222, 223	poimirlem7 36985	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
225	219, 224	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
226	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
227	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑇 ∈ 𝑆)
228	35	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
229		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
230	226, 3, 227, 228, 229	poimirlem7 36985	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑘 − 2))‘𝑛) ≠ ((𝐹‘(𝑘 − 1))‘𝑛)) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
231	225, 230	eqtr3d 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
232	214, 231	jaodan 954	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...((2^nd ‘𝑇) − 1)) ∨ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
233	130, 232	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
234		fvres 6900	. . . 4 ⊢ (𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
235	234	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘) = ((2^nd ‘(1^st ‘𝑈))‘𝑘))
236		fvres 6900	. . . 4 ⊢ (𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
237	236	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
238	233, 235, 237	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘) = (((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))‘𝑘))
239	18, 34, 238	eqfnfvd 7025	1 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 {cab 2701 ≠ wne 2932 ∃*wrmo 3367 {crab 3424 ⦋csb 3885 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ifcif 4520 {csn 4620 {cpr 4622 class class class wbr 5138 ↦ cmpt 5221 × cxp 5664 ↾ cres 5668 “ cima 5669 Fn wfn 6528 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 ℩crio 7356 (class class class)co 7401 ∘_f cof 7661 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8816 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11245 ≤ cle 11246 − cmin 11441 ℕcn 12209 2c2 12264 ℤcz 12555 ℤ_≥cuz 12819 ...cfz 13481 ..^cfzo 13624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625

This theorem is referenced by: poimirlem9 36987

W3C validator