poimirlem8 - Mathbox for Brendan Leahy

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Theorem poimirlem8 38547

Description: Lemma for poimir 38572, establishing that away from the opposite vertex the walks in poimirlem9 38548 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 3640	. . . . . . . . 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 2878	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5	1, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6		xp1st 8033	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8		xp2nd 8034	. . . . . 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 18	. . . . 5 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10		fvex 6898	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
11		f1oeq1 6812	. . . . . 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 3632	. . . . 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 221	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
14		f1ofn 6825	. . . 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 18	. . 3 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁))
16		difss 4072	. . 3 ⊢ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ⊆ (1...𝑁)
17		fnssres 6662	. . 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 598	. 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 3640	. . . . . . . . 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 2878	. . . . . . . 8 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
22	19, 21	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23		xp1st 8033	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25		xp2nd 8034	. . . . . 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 18	. . . . 5 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
27		fvex 6898	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
28		f1oeq1 6812	. . . . . 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 3632	. . . . 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 221	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
31		f1ofn 6825	. . . 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 18	. . 3 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
33		fnssres 6662	. . 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 598	. 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 13711	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
38		poimir.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
39	38	nncnd 12351	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
40		npcan1 11741	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
41	39, 40	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
42	41	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
43	37, 42	eleqtrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
44		fzsplit 13684	. . . . . . . . . 10 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
45	43, 44	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = ((1...((2^nd ‘𝑇) + 1)) ∪ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
46	45	difeq1d 4062	. . . . . . . 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 4236	. . . . . . . . 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 13687	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
49	35, 48	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
50	49	nncnd 12351	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℂ)
51		npcan1 11741	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) ∈ ℂ → (((2^nd ‘𝑇) − 1) + 1) = (2^nd ‘𝑇))
52	50, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) = (2^nd ‘𝑇))
53		nnuz 13004	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
54	49, 53	eleqtrdi 2870	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘1))
55	52, 54	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘1))
56	49	nnzd 12719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℤ)
57		peano2zm 12739	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ ℤ)
58	56, 57	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℤ)
59		uzid 12980	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) − 1) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
60		peano2uz 13028	. . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
62	52, 61	eqeltrrd 2861	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
63		peano2uz 13028	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) → ((2^nd ‘𝑇) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
64	62, 63	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
65		fzsplit2 13683	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1))))
67	52	oveq1d 7435	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)))
68		fzpr 13713	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
69	56, 68	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
70	67, 69	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘𝑇) − 1) + 1)...((2^nd ‘𝑇) + 1)) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
71	70	uneq2d 4167	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...((2^nd ‘𝑇) + 1)) = ((1...((2^nd ‘𝑇) − 1)) ∪ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
73	72	difeq1d 4062	. . . . . . . . . . 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 12350	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
75	74	ltm1d 12249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇))
76	58	zred 12803	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℝ)
77	76, 74	ltnled 11457	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇) ↔ ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1)))
78	75, 77	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
79		elfzle2 13661	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)) → (2^nd ‘𝑇) ≤ ((2^nd ‘𝑇) − 1))
80	78, 79	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ (1...((2^nd ‘𝑇) − 1)))
81		difsn 4760	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) − 1)) ∖ {(2^nd ‘𝑇)}) = (1...((2^nd ‘𝑇) − 1)))
83		peano2re 11483	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℝ → ((2^nd ‘𝑇) + 1) ∈ ℝ)
84	74, 83	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℝ)
85	74	ltp1d 12247	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
86	76, 74, 84, 75, 85	lttrd 11471	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1))
87	76, 84	ltnled 11457	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) − 1) < ((2^nd ‘𝑇) + 1) ↔ ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1)))
88	86, 87	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
89		elfzle2 13661	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)) → ((2^nd ‘𝑇) + 1) ≤ ((2^nd ‘𝑇) − 1))
90	88, 89	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ (1...((2^nd ‘𝑇) − 1)))
91		difsn 4760	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) − 1)) ∖ {((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
93	82, 92	ineq12d 4118	. . . . . . . . . . . 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 4436	. . . . . . . . . . . . 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 4586	. . . . . . . . . . . . . 14 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
96	95	difeq2i 4060	. . . . . . . . . . . . 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 4235	. . . . . . . . . . . . 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 2788	. . . . . . . . . . . 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 4123	. . . . . . . . . . . 12 ⊢ ((1...((2^nd ‘𝑇) − 1)) ∩ (1...((2^nd ‘𝑇) − 1))) = (1...((2^nd ‘𝑇) − 1))
100	93, 98, 99	3eqtr3g 2818	. . . . . . . . . . 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 2795	. . . . . . . . . 10 ⊢ (𝜑 → ((1...((2^nd ‘𝑇) + 1)) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (1...((2^nd ‘𝑇) − 1)))
102		peano2re 11483	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) + 1) ∈ ℝ → (((2^nd ‘𝑇) + 1) + 1) ∈ ℝ)
103	84, 102	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ∈ ℝ)
104	84	ltp1d 12247	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1))
105	74, 84, 103, 85, 104	lttrd 11471	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1))
106	74, 103	ltnled 11457	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇)))
107	105, 106	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
108		elfzle1 13660	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ (2^nd ‘𝑇))
109	107, 108	nsyl 141	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
110		difsn 4760	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {(2^nd ‘𝑇)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
112	84, 103	ltnled 11457	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1) ↔ ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1)))
113	104, 112	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
114		elfzle1 13660	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ ((2^nd ‘𝑇) + 1))
115	113, 114	nsyl 141	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ((2^nd ‘𝑇) + 1) ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
116		difsn 4760	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∖ {((2^nd ‘𝑇) + 1)}) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
118	111, 117	ineq12d 4118	. . . . . . . . . . 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 4060	. . . . . . . . . . . 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 4235	. . . . . . . . . . . 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 2784	. . . . . . . . . . 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 4123	. . . . . . . . . . 11 ⊢ (((((2^nd ‘𝑇) + 1) + 1)...𝑁) ∩ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁)
123	118, 121, 122	3eqtr3g 2818	. . . . . . . . . 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 4168	. . . . . . . . 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 2807	. . . . . . . 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 2795	. . . . . . 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 2846	. . . . . 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 4152	. . . . . 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 290	. . . . 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 482	. . . 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 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
132	131	breq2d 5114	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
133	132	ifbid 4505	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
134	133	csbeq1d 3850	. . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
136		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
137	136	imaeq1d 6050	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
138	137	xpeq1d 5679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
139	136	imaeq1d 6050	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
140	139	xpeq1d 5679	. . . . . . . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . 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 3853	. . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 8033	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
152		elmapi 8869	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
153	151, 152	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
154		elfzoelz 13793	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
155	154	ssriv 3912	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ⊆ ℤ
156		fss 6726	. . . . . . . . . . . . . 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 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
158	38, 149, 157, 30, 35	poimirlem1 38540	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
159	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → 𝑁 ∈ ℕ)
160		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (2^nd ‘𝑡) = (2^nd ‘𝑈))
161	160	breq2d 5114	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑈)))
162	161	ifbid 4505	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)))
163	162	csbeq1d 3850	. . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑈 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑈)))
165		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑈 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑈)))
166	165	imaeq1d 6050	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)))
167	166	xpeq1d 5679	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}))
168	165	imaeq1d 6050	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
169	168	xpeq1d 5679	. . . . . . . . . . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . . . . 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 3853	. . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 8033	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
182		elmapi 8869	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
183	181, 182	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
184		fss 6726	. . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
186	185	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶ℤ)
187	13	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
188	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
189		xp2nd 8034	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (0...𝑁))
191		eldifsn 4747	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}) ↔ ((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)))
192	191	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘𝑈) ∈ (0...𝑁) ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
193	190, 192	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑈) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑈) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
194	159, 179, 186, 187, 188, 193	poimirlem2 38541	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 2973	. . . . . . . . . . . 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 16	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘𝑈) = (2^nd ‘𝑇))
198	197	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘𝑈) − 1) = ((2^nd ‘𝑇) − 1))
199	198	oveq2d 7436	. . . . . . . . 9 ⊢ (𝜑 → (1...((2^nd ‘𝑈) − 1)) = (1...((2^nd ‘𝑇) − 1)))
200	199	eleq2d 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((2^nd ‘𝑈) − 1)) ↔ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))))
201	200	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
202	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑁 ∈ ℕ)
203	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑈 ∈ 𝑆)
204	197, 35	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
205	204	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
206		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑈) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑈) − 1)))
207	202, 3, 203, 205, 206	poimirlem6 38545	. . . . . . 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 603	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑁 ∈ ℕ)
210	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑇 ∈ 𝑆)
211	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
212		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → 𝑘 ∈ (1...((2^nd ‘𝑇) − 1)))
213	209, 3, 210, 211, 212	poimirlem6 38545	. . . . . 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 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((2^nd ‘𝑇) − 1))) → ((2^nd ‘(1^st ‘𝑈))‘𝑘) = ((2^nd ‘(1^st ‘𝑇))‘𝑘))
215	197	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘𝑈) + 1) = ((2^nd ‘𝑇) + 1))
216	215	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘𝑈) + 1) + 1) = (((2^nd ‘𝑇) + 1) + 1))
217	216	oveq1d 7435	. . . . . . . . 9 ⊢ (𝜑 → ((((2^nd ‘𝑈) + 1) + 1)...𝑁) = ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
218	217	eleq2d 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁) ↔ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)))
219	218	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
220	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
221	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑈 ∈ 𝑆)
222	204	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → (2^nd ‘𝑈) ∈ (1...(𝑁 − 1)))
223		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑈) + 1) + 1)...𝑁))
224	220, 3, 221, 222, 223	poimirlem7 38546	. . . . . . 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 603	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑁 ∈ ℕ)
227	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑇 ∈ 𝑆)
228	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
229		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁)) → 𝑘 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
230	226, 3, 227, 228, 229	poimirlem7 38546	. . . . . 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 2797	. . . . 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 972	. . . 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 603	. . 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 6904	. . . 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 487	. . 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 6904	. . . 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 487	. . 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 2805	. 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 7032	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 {cab 2738 ≠ wne 2955 ∃*wrmo 3364 {crab 3412 ⦋csb 3846 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ifcif 4481 {csn 4583 {cpr 4585 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ↾ cres 5652 “ cima 5653 Fn wfn 6533 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 ℩crio 7376 (class class class)co 7420 ∘_f cof 7691 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8847 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 < clt 11343 ≤ cle 11344 − cmin 11541 ℕcn 12335 2c2 12397 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 ..^cfzo 13788

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789

This theorem is used by: poimirlem9 38548

W3C validator