poimirlem8 - Mathbox for Brendan Leahy

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Theorem poimirlem8 36485

Description: Lemma for poimir 36510, establishing that away from the opposite vertex the walks in poimirlem9 36486 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 3677	. . . . . . . . 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 2852	. . . . . . . 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 8004	. . . . . . 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 8005	. . . . . 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 6902	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
11		f1oeq1 6819	. . . . . 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 3668	. . . . 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 6832	. . . 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 4131	. . 3 ⊢ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)
17		fnssres 6671	. . 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 587	. 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 3677	. . . . . . . . 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 2852	. . . . . . . 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 8004	. . . . . . 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 8005	. . . . . 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 6902	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
28		f1oeq1 6819	. . . . . 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 3668	. . . . 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 6832	. . . 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 6671	. . 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 587	. 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 7422	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
43	37, 42	eleqtrd 2836	. . . . . . . . . 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 4121	. . . . . . . 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 4280	. . . . . . . . 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 2844	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘1))
55	52, 54	eqeltrd 2834	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))))
67	52	oveq1d 7421	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
71	70	uneq2d 4163	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
73	72	difeq1d 4121	. . . . . . . . . . 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 4801	. . . . . . . . . . . . . 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 4801	. . . . . . . . . . . . . 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 4213	. . . . . . . . . . . 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 4480	. . . . . . . . . . . . 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 4631	. . . . . . . . . . . . . 14 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
96	95	difeq2i 4119	. . . . . . . . . . . . 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 4279	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 4218	. . . . . . . . . . . 12 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∩ (1...((2^nd ‘𝑇) − 1))) = (1...((2^nd ‘𝑇) − 1))
100	93, 98, 99	3eqtr3g 2796	. . . . . . . . . . 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 2773	. . . . . . . . . 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 4801	. . . . . . . . . . . . 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 4801	. . . . . . . . . . . . 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 4213	. . . . . . . . . . 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 4119	. . . . . . . . . . . 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 4279	. . . . . . . . . . . 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 2762	. . . . . . . . . . 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 4218	. . . . . . . . . . 11 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁)
123	118, 121, 122	3eqtr3g 2796	. . . . . . . . . 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 4164	. . . . . . . . 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 2785	. . . . . . . 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 2773	. . . . . . 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 2820	. . . . . 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 4148	. . . . . 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 478	. . . 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 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
132	131	breq2d 5160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
133	132	ifbid 4551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
134	133	csbeq1d 3897	. . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
136		2fveq3 6894	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
137	136	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
138	137	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
139	136	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
140	139	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . 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 4164	. . . . . . . . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . 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 5250	. . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . 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 8004	. . . . . . . . . . . . . . . 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 8840	. . . . . . . . . . . . . . 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 3986	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ⊆ ℤ
156		fss 6732	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
158	38, 149, 157, 30, 35	poimirlem1 36478	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
159	38	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → 𝑁 ∈ ℕ)
160		fveq2 6889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (2^nd ‘𝑡) = (2^nd ‘𝑈))
161	160	breq2d 5160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑈)))
162	161	ifbid 4551	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)))
163	162	csbeq1d 3897	. . . . . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑈)))
165		2fveq3 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑈 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑈)))
166	165	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)))
167	166	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}))
168	165	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
169	168	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . 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 4164	. . . . . . . . . . . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . 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 5250	. . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 8004	. . . . . . . . . . . . . . . . . . 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 8840	. . . . . . . . . . . . . . . . . 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 6732	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
186	185	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
187	13	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
188	35	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
189		xp2nd 8005	. . . . . . . . . . . . . . . . 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 4790	. . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
194	159, 179, 186, 187, 188, 193	poimirlem2 36479	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 2959	. . . . . . . . . . . 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 7421	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑈) − 1) = ((2^nd ‘𝑇) − 1))
199	198	oveq2d 7422	. . . . . . . . 9 ⊢ (𝜑 → (1...((2^nd ‘𝑈) − 1)) = (1...((2^nd ‘𝑇) − 1)))
200	199	eleq2d 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((2^nd ‘𝑈) − 1)) ↔ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))))
201	200	biimpar 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
202	38	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑁 ∈ ℕ)
203	1	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑈 ∈ 𝑆)
204	197, 35	eqeltrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
205	204	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
206		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
207	202, 3, 203, 205, 206	poimirlem6 36483	. . . . . . 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 592	. . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑁 ∈ ℕ)
210	19	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑇 ∈ 𝑆)
211	35	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
212		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑇) − 1)))
213	209, 3, 210, 211, 212	poimirlem6 36483	. . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
215	197	oveq1d 7421	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑈) + 1) = ((2^nd ‘𝑇) + 1))
216	215	oveq1d 7421	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘𝑈) + 1) + 1) = (((2^nd ‘𝑇) + 1) + 1))
217	216	oveq1d 7421	. . . . . . . . 9 ⊢ (𝜑 → ((((2^nd ‘𝑈) + 1) + 1)...𝑁) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
218	217	eleq2d 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
219	218	biimpar 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
220	38	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
221	1	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑈 ∈ 𝑆)
222	204	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
223		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
224	220, 3, 221, 222, 223	poimirlem7 36484	. . . . . . 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 592	. . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
227	19	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑇 ∈ 𝑆)
228	35	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
229		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
230	226, 3, 227, 228, 229	poimirlem7 36484	. . . . . 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 2775	. . . . 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 957	. . . 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 592	. . 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 6908	. . . 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 483	. . 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 6908	. . . 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 483	. . 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 2783	. 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 7033	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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∃*wrmo 3376 {crab 3433 ⦋csb 3893 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ifcif 4528 {csn 4628 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 “ cima 5679 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 ℩crio 7361 (class class class)co 7406 ∘_f cof 7665 1^st c1st 7970 2^nd c2nd 7971 ↑_m cmap 8817 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 < 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 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 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 36486

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