poimirlem8 - Mathbox for Brendan Leahy

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Theorem poimirlem8 34904

Description: Lemma for poimir 34929, establishing that away from the opposite vertex the walks in poimirlem9 34905 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 3678	. . . . . . . . 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 2934	. . . . . . . 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 7724	. . . . . . 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 7725	. . . . . 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 6686	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
11		f1oeq1 6607	. . . . . 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 3670	. . . . 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 220	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14		f1ofn 6619	. . . 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 4111	. . 3 ⊢ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)
17		fnssres 6473	. . 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 588	. 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 3678	. . . . . . . . 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 2934	. . . . . . . 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 7724	. . . . . . 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 7725	. . . . . 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 6686	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
28		f1oeq1 6607	. . . . . 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 3670	. . . . 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 220	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31		f1ofn 6619	. . . 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 6473	. . 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 588	. 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 12963	. . . . . . . . . . . 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 11657	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
40		npcan1 11068	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
42	41	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
43	37, 42	eleqtrd 2918	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
44		fzsplit 12936	. . . . . . . . . 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 4101	. . . . . . . 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 4260	. . . . . . . . 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 12939	. . . . . . . . . . . . . . . . . 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 11657	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℂ)
51		npcan1 11068	. . . . . . . . . . . . . . . 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 12284	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
54	49, 53	eleqtrdi 2926	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘1))
55	52, 54	eqeltrd 2916	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘1))
56	49	nnzd 12089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℤ)
57		peano2zm 12028	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ ℤ)
58	56, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℤ)
59		uzid 12261	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) − 1) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
60		peano2uz 12304	. . . . . . . . . . . . . . . . 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 2917	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
63		peano2uz 12304	. . . . . . . . . . . . . . 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 12935	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))))
67	52	oveq1d 7174	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)))
68		fzpr 12965	. . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
71	70	uneq2d 4142	. . . . . . . . . . . . 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 2859	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
73	72	difeq1d 4101	. . . . . . . . . . 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 11656	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
75	74	ltm1d 11575	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇))
76	58	zred 12090	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℝ)
77	76, 74	ltnled 10790	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇) ↔ ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1)))
78	75, 77	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
79		elfzle2 12914	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)) → (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
80	78, 79	nsyl 142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)))
81		difsn 4734	. . . . . . . . . . . . . 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 10816	. . . . . . . . . . . . . . . . . 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 11573	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
86	76, 74, 84, 75, 85	lttrd 10804	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1))
87	76, 84	ltnled 10790	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1)))
88	86, 87	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
89		elfzle2 12914	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)) → ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
90	88, 89	nsyl 142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)))
91		difsn 4734	. . . . . . . . . . . . . 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 4193	. . . . . . . . . . . 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 4432	. . . . . . . . . . . . 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 4573	. . . . . . . . . . . . . 14 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
96	95	difeq2i 4099	. . . . . . . . . . . . 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 4259	. . . . . . . . . . . . 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 2852	. . . . . . . . . . . 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 4198	. . . . . . . . . . . 12 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∩ (1...((2^nd ‘𝑇) − 1))) = (1...((2^nd ‘𝑇) − 1))
100	93, 98, 99	3eqtr3g 2882	. . . . . . . . . . 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 2859	. . . . . . . . . 10 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
102		peano2re 10816	. . . . . . . . . . . . . . . . 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 11573	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1))
105	74, 84, 103, 85, 104	lttrd 10804	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1))
106	74, 103	ltnled 10790	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇)))
107	105, 106	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
108		elfzle1 12913	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
109	107, 108	nsyl 142	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
110		difsn 4734	. . . . . . . . . . . . 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 10790	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
113	104, 112	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
114		elfzle1 12913	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
115	113, 114	nsyl 142	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
116		difsn 4734	. . . . . . . . . . . . 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 4193	. . . . . . . . . . 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 4099	. . . . . . . . . . . 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 4259	. . . . . . . . . . . 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 2848	. . . . . . . . . . 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 4198	. . . . . . . . . . 11 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁)
123	118, 121, 122	3eqtr3g 2882	. . . . . . . . . 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 4143	. . . . . . . . 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	syl5eq 2871	. . . . . . . 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 2859	. . . . . . 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 2901	. . . . . 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 4128	. . . . . 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	syl6bb 289	. . . . 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 479	. . . 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 6673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
132	131	breq2d 5081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
133	132	ifbid 4492	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
134	133	csbeq1d 3890	. . . . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
136		2fveq3 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
137	136	imaeq1d 5931	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
138	137	xpeq1d 5587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
139	136	imaeq1d 5931	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
140	139	xpeq1d 5587	. . . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . . 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 7177	. . . . . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . 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 3686	. . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . 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 7724	. . . . . . . . . . . . . . . 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 8431	. . . . . . . . . . . . . . 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 13041	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155	154	ssriv 3974	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ⊆ ℤ
156		fss 6530	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
158	38, 149, 157, 30, 35	poimirlem1 34897	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
159	38	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → 𝑁 ∈ ℕ)
160		fveq2 6673	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (2^nd ‘𝑡) = (2^nd ‘𝑈))
161	160	breq2d 5081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑈)))
162	161	ifbid 4492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)))
163	162	csbeq1d 3890	. . . . . . . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑈)))
165		2fveq3 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑈 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑈)))
166	165	imaeq1d 5931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)))
167	166	xpeq1d 5587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}))
168	165	imaeq1d 5931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
169	168	xpeq1d 5587	. . . . . . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . . . . . 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 7177	. . . . . . . . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . . . 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 3686	. . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 7724	. . . . . . . . . . . . . . . . . . 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 8431	. . . . . . . . . . . . . . . . . 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 6530	. . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
186	185	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
187	13	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
188	35	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
189		xp2nd 7725	. . . . . . . . . . . . . . . . 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 4722	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}) ↔ ((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)))
192	191	biimpri 230	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
193	190, 192	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
194	159, 179, 186, 187, 188, 193	poimirlem2 34898	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 3037	. . . . . . . . . . . 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 7174	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑈) − 1) = ((2^nd ‘𝑇) − 1))
199	198	oveq2d 7175	. . . . . . . . 9 ⊢ (𝜑 → (1...((2^nd ‘𝑈) − 1)) = (1...((2^nd ‘𝑇) − 1)))
200	199	eleq2d 2901	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((2^nd ‘𝑈) − 1)) ↔ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))))
201	200	biimpar 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
202	38	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑁 ∈ ℕ)
203	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑈 ∈ 𝑆)
204	197, 35	eqeltrd 2916	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
205	204	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
206		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
207	202, 3, 203, 205, 206	poimirlem6 34902	. . . . . . 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 593	. . . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑁 ∈ ℕ)
210	19	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑇 ∈ 𝑆)
211	35	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
212		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑇) − 1)))
213	209, 3, 210, 211, 212	poimirlem6 34902	. . . . . 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 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
215	197	oveq1d 7174	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑈) + 1) = ((2^nd ‘𝑇) + 1))
216	215	oveq1d 7174	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘𝑈) + 1) + 1) = (((2^nd ‘𝑇) + 1) + 1))
217	216	oveq1d 7174	. . . . . . . . 9 ⊢ (𝜑 → ((((2^nd ‘𝑈) + 1) + 1)...𝑁) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
218	217	eleq2d 2901	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
219	218	biimpar 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
220	38	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
221	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑈 ∈ 𝑆)
222	204	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
223		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
224	220, 3, 221, 222, 223	poimirlem7 34903	. . . . . . 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 593	. . . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
227	19	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑇 ∈ 𝑆)
228	35	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
229		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
230	226, 3, 227, 228, 229	poimirlem7 34903	. . . . . 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 2861	. . . . 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 593	. . 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 6692	. . . 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 484	. . 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 6692	. . . 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 484	. . 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 2869	. 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 6808	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 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 {cab 2802 ≠ wne 3019 ∃*wrmo 3144 {crab 3145 ⦋csb 3886 ∖ cdif 3936 ∪ cun 3937 ∩ cin 3938 ⊆ wss 3939 ifcif 4470 {csn 4570 {cpr 4572 class class class wbr 5069 ↦ cmpt 5149 × cxp 5556 ↾ cres 5560 “ cima 5561 Fn wfn 6353 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 ℩crio 7116 (class class class)co 7159 ∘_f cof 7410 1^st c1st 7690 2^nd c2nd 7691 ↑_m cmap 8409 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 < clt 10678 ≤ cle 10679 − cmin 10873 ℕcn 11641 2c2 11695 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ..^cfzo 13036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037

This theorem is referenced by: poimirlem9 34905

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