poimirlem22 - Mathbox for Brendan Leahy

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Theorem poimirlem22 33561

Description: Lemma for poimir 33572, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (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}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
poimirlem22.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem22.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
poimirlem22.4	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)

**Assertion**
Ref	Expression
poimirlem22	⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Distinct variable groups: 𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 𝜑,𝑗,𝑛,𝑦 𝑗,𝐹,𝑛,𝑦 𝑗,𝑁,𝑛,𝑦 𝑇,𝑗,𝑛,𝑦 𝜑,𝑝,𝑡 𝑓,𝐾,𝑗,𝑛,𝑝,𝑡 𝑓,𝑁,𝑝,𝑡 𝑇,𝑓,𝑝 𝜑,𝑧 𝑓,𝐹,𝑝,𝑡,𝑧 𝑧,𝐾 𝑧,𝑁 𝑡,𝑇,𝑧 𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝑆(𝑓) 𝐾(𝑦)

**Proof of Theorem poimirlem22**
Step	Hyp	Ref	Expression
1		poimir.0	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
3		poimirlem22.s	. . . 4 ⊢ 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		poimirlem22.1	. . . . 5 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
6		poimirlem22.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → 𝑇 ∈ 𝑆)
8		simpr 476	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
9	2, 3, 5, 7, 8	poimirlem15 33554	. . 3 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ 𝑆)
10		fveq2 6229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
11	10	breq2d 4697	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
12	11	ifbid 4141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
13	12	csbeq1d 3573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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}))))
14		fveq2 6229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (1^st ‘𝑡) = (1^st ‘𝑇))
15	14	fveq2d 6233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
16	14	fveq2d 6233	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
17	16	imaeq1d 5500	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
18	17	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
19	16	imaeq1d 5500	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
20	19	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
21	18, 20	uneq12d 3801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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})))
22	15, 21	oveq12d 6708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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}))))
23	22	csbeq2dv 4025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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}))))
24	13, 23	eqtrd 2685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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}))))
25	24	mpteq2dv 4778	. . . . . . . . . . . . . . . . . . 19 ⊢ (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})))))
26	25	eqeq2d 2661	. . . . . . . . . . . . . . . . . 18 ⊢ (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}))))))
27	26, 3	elrab2 3399	. . . . . . . . . . . . . . . . 17 ⊢ (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}))))))
28	27	simprbi 479	. . . . . . . . . . . . . . . 16 ⊢ (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})))))
29	6, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (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})))))
30	29	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → 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})))))
31		elrabi 3391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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)))
32	31, 3	eleq2s 2748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33	6, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
34		xp1st 7242	. . . . . . . . . . . . . . . . . . 19 ⊢ (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)}))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
36		xp1st 7242	. . . . . . . . . . . . . . . . . 18 ⊢ ((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)))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
38		elmapi 7921	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
40		elfzoelz 12509	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
41	40	ssriv 3640	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
42		fss 6094	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
43	39, 41, 42	sylancl 695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
44	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶ℤ)
45		xp2nd 7243	. . . . . . . . . . . . . . . . 17 ⊢ ((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)})
46	35, 45	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
47		fvex 6239	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
48		f1oeq1 6165	. . . . . . . . . . . . . . . . 17 ⊢ (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)))
49	47, 48	elab 3382	. . . . . . . . . . . . . . . 16 ⊢ ((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))
50	46, 49	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
51	50	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
52	2, 30, 44, 51, 8	poimirlem1 33540	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
53	52	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
54	1	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → 𝑁 ∈ ℕ)
55		fveq2 6229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → (2^nd ‘𝑡) = (2^nd ‘𝑧))
56	55	breq2d 4697	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑧)))
57	56	ifbid 4141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)))
58	57	csbeq1d 3573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → ⦋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 ‘𝑧), 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}))))
59		fveq2 6229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → (1^st ‘𝑡) = (1^st ‘𝑧))
60	59	fveq2d 6233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑧)))
61	59	fveq2d 6233	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑧 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑧)))
62	61	imaeq1d 5500	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)))
63	62	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}))
64	61	imaeq1d 5500	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
65	64	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
66	63, 65	uneq12d 3801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))
67	60, 66	oveq12d 6708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → ((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 ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
68	67	csbeq2dv 4025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → ⦋if(𝑦 < (2^nd ‘𝑧), 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 ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
69	58, 68	eqtrd 2685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑧 → ⦋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 ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70	69	mpteq2dv 4778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑧 → (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 ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
71	70	eqeq2d 2661	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑧 → (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 ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
72	71, 3	elrab2 3399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 ↔ (𝑧 ∈ ((((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 ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
73	72	simprbi 479	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
74	73	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
75		elrabi 3391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {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}))))} → 𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
76	75, 3	eleq2s 2748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
77		xp1st 7242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
79		xp1st 7242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑧)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘(1^st ‘𝑧)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
81		elmapi 7921	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(1^st ‘𝑧)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)) → (1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
82	80, 81	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
83		fss 6094	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶ℤ)
84	82, 41, 83	sylancl 695	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶ℤ)
85	84	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → (1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶ℤ)
86		xp2nd 7243	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
87	78, 86	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
88		fvex 6239	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘(1^st ‘𝑧)) ∈ V
89		f1oeq1 6165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
90	88, 89	elab 3382	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
91	87, 90	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
92	91	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
93		simpllr 815	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
94		xp2nd 7243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2^nd ‘𝑧) ∈ (0...𝑁))
95	76, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘𝑧) ∈ (0...𝑁))
96	95	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (2^nd ‘𝑧) ∈ (0...𝑁))
97		eldifsn 4350	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑧) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}) ↔ ((2^nd ‘𝑧) ∈ (0...𝑁) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)))
98	97	biimpri 218	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘𝑧) ∈ (0...𝑁) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑧) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
99	96, 98	sylan 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → (2^nd ‘𝑧) ∈ ((0...𝑁) ∖ {(2^nd ‘𝑇)}))
100	54, 74, 85, 92, 93, 99	poimirlem2 33541	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) ≠ (2^nd ‘𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛))
101	100	ex 449	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → ((2^nd ‘𝑧) ≠ (2^nd ‘𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛)))
102	101	necon1bd 2841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2^nd ‘𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2^nd ‘𝑇))‘𝑛) → (2^nd ‘𝑧) = (2^nd ‘𝑇)))
103	53, 102	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (2^nd ‘𝑧) = (2^nd ‘𝑇))
104		eleq1 2718	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑧) = (2^nd ‘𝑇) → ((2^nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))))
105	104	biimparc 503	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → (2^nd ‘𝑧) ∈ (1...(𝑁 − 1)))
106	105	anim2i 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))) → (𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))))
107	106	anassrs 681	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → (𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))))
108	73	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
109		breq1 4688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 0 → (𝑦 < (2^nd ‘𝑧) ↔ 0 < (2^nd ‘𝑧)))
110		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 0 → 𝑦 = 0)
111	109, 110	ifbieq1d 4142	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 0 → if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) = if(0 < (2^nd ‘𝑧), 0, (𝑦 + 1)))
112		elfznn 12408	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑧) ∈ ℕ)
113	112	nngt0d 11102	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑧) ∈ (1...(𝑁 − 1)) → 0 < (2^nd ‘𝑧))
114	113	iftrued 4127	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑧) ∈ (1...(𝑁 − 1)) → if(0 < (2^nd ‘𝑧), 0, (𝑦 + 1)) = 0)
115	114	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → if(0 < (2^nd ‘𝑧), 0, (𝑦 + 1)) = 0)
116	111, 115	sylan9eqr 2707	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) = 0)
117	116	csbeq1d 3573	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) → ⦋if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋0 / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
118		c0ex 10072	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
119		oveq2 6698	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
120		fz10 12400	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...0) = ∅
121	119, 120	syl6eq 2701	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
122	121	imaeq2d 5501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑧)) “ ∅))
123	122	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 0 → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}))
124		oveq1 6697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
125		0p1e1 11170	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 + 1) = 1
126	124, 125	syl6eq 2701	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
127	126	oveq1d 6705	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
128	127	imaeq2d 5501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → ((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)))
129	128	xpeq1d 5172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 0 → (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}))
130	123, 129	uneq12d 3801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 0 → ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})))
131		ima0 5516	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ∅
132	131	xpeq1i 5169	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}) = (∅ × {1})
133		0xp 5233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {1}) = ∅
134	132, 133	eqtri 2673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}) = ∅
135	134	uneq1i 3796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}))
136		uncom 3790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∪ (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}) ∪ ∅)
137		un0 4000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})
138	135, 136, 137	3eqtri 2677	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((2^nd ‘(1^st ‘𝑧)) “ ∅) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})
139	130, 138	syl6eq 2701	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 0 → ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}))
140	139	oveq2d 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 0 → ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})))
141	118, 140	csbie 3592	. . . . . . . . . . . . . . . . 17 ⊢ ⦋0 / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}))
142		f1ofo 6182	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–onto→(1...𝑁))
143	91, 142	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–onto→(1...𝑁))
144		foima 6158	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) = (1...𝑁))
145	143, 144	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑆 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) = (1...𝑁))
146	145	xpeq1d 5172	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
147	146	oveq2d 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})) = ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((1...𝑁) × {0})))
148		ovexd 6720	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → (1...𝑁) ∈ V)
149		ffn 6083	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1^st ‘(1^st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) → (1^st ‘(1^st ‘𝑧)) Fn (1...𝑁))
150	82, 149	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘(1^st ‘𝑧)) Fn (1...𝑁))
151		fnconstg 6131	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
152	118, 151	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → ((1...𝑁) × {0}) Fn (1...𝑁))
153		eqidd 2652	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑧))‘𝑛) = ((1^st ‘(1^st ‘𝑧))‘𝑛))
154	118	fvconst2 6510	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
155	154	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
156	82	ffvelrnda 6399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑧))‘𝑛) ∈ (0..^𝐾))
157		elfzonn0 12552	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘(1^st ‘𝑧))‘𝑛) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑧))‘𝑛) ∈ ℕ₀)
158	156, 157	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑧))‘𝑛) ∈ ℕ₀)
159	158	nn0cnd 11391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑧))‘𝑛) ∈ ℂ)
160	159	addid1d 10274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑧))‘𝑛) + 0) = ((1^st ‘(1^st ‘𝑧))‘𝑛))
161	148, 150, 152, 150, 153, 155, 160	offveq 6960	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((1...𝑁) × {0})) = (1^st ‘(1^st ‘𝑧)))
162	147, 161	eqtrd 2685	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → ((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑁)) × {0})) = (1^st ‘(1^st ‘𝑧)))
163	141, 162	syl5eq 2697	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → ⦋0 / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1^st ‘(1^st ‘𝑧)))
164	163	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) → ⦋0 / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1^st ‘(1^st ‘𝑧)))
165	117, 164	eqtrd 2685	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑦 = 0) → ⦋if(𝑦 < (2^nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑧)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1^st ‘(1^st ‘𝑧)))
166		nnm1nn0 11372	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
167	1, 166	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
168		0elfz 12475	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ₀ → 0 ∈ (0...(𝑁 − 1)))
169	167, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0...(𝑁 − 1)))
170	169	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → 0 ∈ (0...(𝑁 − 1)))
171		fvexd 6241	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (1^st ‘(1^st ‘𝑧)) ∈ V)
172	108, 165, 170, 171	fvmptd 6327	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧)))
173	107, 172	sylan 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) ∧ 𝑧 ∈ 𝑆) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧)))
174	173	an32s 863	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧)))
175	103, 174	mpdan 703	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧)))
176		fveq2 6229	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑇 → (2^nd ‘𝑧) = (2^nd ‘𝑇))
177	176	eleq1d 2715	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑇 → ((2^nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))))
178	177	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑇 → ((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) ↔ (𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))))
179		fveq2 6229	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑇 → (1^st ‘𝑧) = (1^st ‘𝑇))
180	179	fveq2d 6233	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑇 → (1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)))
181	180	eqeq2d 2661	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑇 → ((𝐹‘0) = (1^st ‘(1^st ‘𝑧)) ↔ (𝐹‘0) = (1^st ‘(1^st ‘𝑇))))
182	178, 181	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑇 → (((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧))) ↔ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1^st ‘(1^st ‘𝑇)))))
183	172	expcom 450	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → ((𝜑 ∧ (2^nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1^st ‘(1^st ‘𝑧))))
184	182, 183	vtoclga 3303	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ 𝑆 → ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1^st ‘(1^st ‘𝑇))))
185	7, 184	mpcom 38	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1^st ‘(1^st ‘𝑇)))
186	185	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝐹‘0) = (1^st ‘(1^st ‘𝑇)))
187	175, 186	eqtr3d 2687	. . . . . . . . 9 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)))
188	187	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)))
189	1	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → 𝑁 ∈ ℕ)
190	6	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → 𝑇 ∈ 𝑆)
191		simpllr 815	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
192		simplr 807	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → 𝑧 ∈ 𝑆)
193	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
194		xpopth 7251	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑇))) ↔ (1^st ‘𝑧) = (1^st ‘𝑇)))
195	78, 193, 194	syl2anr 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑇))) ↔ (1^st ‘𝑧) = (1^st ‘𝑇)))
196	33	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
197		xpopth 7251	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1^st ‘𝑧) = (1^st ‘𝑇) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) ↔ 𝑧 = 𝑇))
198	197	biimpd 219	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1^st ‘𝑧) = (1^st ‘𝑇) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → 𝑧 = 𝑇))
199	76, 196, 198	syl2anr 494	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (((1^st ‘𝑧) = (1^st ‘𝑇) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → 𝑧 = 𝑇))
200	103, 199	mpan2d 710	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → ((1^st ‘𝑧) = (1^st ‘𝑇) → 𝑧 = 𝑇))
201	195, 200	sylbid 230	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑇))) → 𝑧 = 𝑇))
202	187, 201	mpand 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → ((2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑇)) → 𝑧 = 𝑇))
203	202	necon3d 2844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2^nd ‘(1^st ‘𝑧)) ≠ (2^nd ‘(1^st ‘𝑇))))
204	203	imp 444	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (2^nd ‘(1^st ‘𝑧)) ≠ (2^nd ‘(1^st ‘𝑇)))
205	189, 3, 190, 191, 192, 204	poimirlem9 33548	. . . . . . . 8 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
206	103	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (2^nd ‘𝑧) = (2^nd ‘𝑇))
207	188, 205, 206	jca31 556	. . . . . . 7 ⊢ ((((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑧 ≠ 𝑇) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)))
208	207	ex 449	. . . . . 6 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
209		simplr 807	. . . . . . . 8 ⊢ ((((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
210		elfznn 12408	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
211	210	nnred 11073	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℝ)
212	211	ltp1d 10992	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
213	211, 212	ltned 10211	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1))
214	213	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1))
215		fveq1 6228	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)))
216		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (2^nd ‘𝑇) ∈ ℝ)
217		ltp1 10899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
218	216, 217	ltned 10211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1))
219		fvex 6239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2^nd ‘𝑇) ∈ V
220		ovex 6718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑇) + 1) ∈ V
221	219, 220, 220, 219	fpr 6461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
222	218, 221	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑇) ∈ ℝ → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
223		f1oi 6212	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
224		f1of 6175	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})⟶((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
225	223, 224	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})⟶((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
226		disjdif 4073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅
227		fun 6104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})⟶((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ∧ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))⟶({((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
228	226, 227	mpan2 707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})):((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})⟶((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))⟶({((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
229	222, 225, 228	sylancl 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑇) ∈ ℝ → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))⟶({((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
230	219	prid1 4329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2^nd ‘𝑇) ∈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}
231		elun1 3813	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑇) ∈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} → (2^nd ‘𝑇) ∈ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
232	230, 231	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (2^nd ‘𝑇) ∈ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
233		fvco3 6314	. . . . . . . . . . . . . . . . . . 19 ⊢ ((({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))):({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))⟶({((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) ∧ (2^nd ‘𝑇) ∈ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘(({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇))))
234	229, 232, 233	sylancl 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘(({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇))))
235		ffn 6083	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
236	222, 235	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑇) ∈ ℝ → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
237		fnresi 6046	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
238	226, 230	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅ ∧ (2^nd ‘𝑇) ∈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
239		fvun1 6308	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ (({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ∅ ∧ (2^nd ‘𝑇) ∈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇)) = ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}‘(2^nd ‘𝑇)))
240	237, 238, 239	mp3an23 1456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} Fn {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇)) = ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}‘(2^nd ‘𝑇)))
241	236, 240	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇)) = ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}‘(2^nd ‘𝑇)))
242	219, 220	fvpr1 6497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}‘(2^nd ‘𝑇)) = ((2^nd ‘𝑇) + 1))
243	218, 242	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑇) ∈ ℝ → ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}‘(2^nd ‘𝑇)) = ((2^nd ‘𝑇) + 1))
244	241, 243	eqtrd 2685	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇)) = ((2^nd ‘𝑇) + 1))
245	244	fveq2d 6233	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑇) ∈ ℝ → ((2^nd ‘(1^st ‘𝑇))‘(({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))‘(2^nd ‘𝑇))) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)))
246	234, 245	eqtrd 2685	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑇) ∈ ℝ → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)))
247	211, 246	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)))
248	247	eqeq2d 2661	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) ↔ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))))
249	248	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) ↔ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))))
250		f1of1 6174	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
251	50, 250	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
252	251	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
253	1	nncnd 11074	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
254		npcan1 10493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
255	253, 254	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
256	167	nn0zd 11518	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
257		uzid 11740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
258	256, 257	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
259		peano2uz 11779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
260	258, 259	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
261	255, 260	eqeltrrd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
262		fzss2 12419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
263	261, 262	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
264	263	sselda 3636	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘𝑇) ∈ (1...𝑁))
265		fzp1elp1 12432	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
266	265	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
267	255	oveq2d 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
268	267	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (1...((𝑁 − 1) + 1)) = (1...𝑁))
269	266, 268	eleqtrd 2732	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
270		f1veqaeq 6554	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ ((2^nd ‘𝑇) ∈ (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))) → (((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) → (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1)))
271	252, 264, 269, 270	syl12anc 1364	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) → (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1)))
272	249, 271	sylbid 230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) = (((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))‘(2^nd ‘𝑇)) → (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1)))
273	215, 272	syl5 34	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1)))
274	273	necon3d 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) → (2^nd ‘(1^st ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))))
275	214, 274	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
276	179	fveq2d 6233	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑇)))
277	276	neeq1d 2882	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → ((2^nd ‘(1^st ‘𝑧)) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) ↔ (2^nd ‘(1^st ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))))
278	275, 277	syl5ibrcom 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → (𝑧 = 𝑇 → (2^nd ‘(1^st ‘𝑧)) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))))
279	278	necon2d 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) → 𝑧 ≠ 𝑇))
280	209, 279	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → 𝑧 ≠ 𝑇))
281	280	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → ((((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) → 𝑧 ≠ 𝑇))
282	208, 281	impbid 202	. . . . 5 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 ↔ (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
283		eqop 7252	. . . . . . . 8 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ↔ ((1^st ‘𝑧) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
284		eqop 7252	. . . . . . . . . 10 ⊢ ((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((1^st ‘𝑧) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ↔ ((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))))
285	77, 284	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((1^st ‘𝑧) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ↔ ((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))))
286	285	anbi1d 741	. . . . . . . 8 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((1^st ‘𝑧) = ⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩ ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇)) ↔ (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
287	283, 286	bitrd 268	. . . . . . 7 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ↔ (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
288	76, 287	syl 17	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → (𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ↔ (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
289	288	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ↔ (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑇)) ∧ (2^nd ‘(1^st ‘𝑧)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑇))))
290	282, 289	bitr4d 271	. . . 4 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 ↔ 𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩))
291	290	ralrimiva 2995	. . 3 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 ↔ 𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩))
292		reu6i 3430	. . 3 ⊢ ((⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩ ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 ↔ 𝑧 = ⟨⟨(1^st ‘(1^st ‘𝑇)), ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))⟩, (2^nd ‘𝑇)⟩)) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
293	9, 291, 292	syl2anc 694	. 2 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
294		xp2nd 7243	. . . . . . 7 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2^nd ‘𝑇) ∈ (0...𝑁))
295	33, 294	syl 17	. . . . . 6 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (0...𝑁))
296	295	biantrurd 528	. . . . 5 ⊢ (𝜑 → (¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2^nd ‘𝑇) ∈ (0...𝑁) ∧ ¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))))
297	1	nnnn0d 11389	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
298		nn0uz 11760	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
299	297, 298	syl6eleq 2740	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
300		fzpred 12427	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
301	299, 300	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
302	125	oveq1i 6700	. . . . . . . . . . 11 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
303	302	uneq2i 3797	. . . . . . . . . 10 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
304	301, 303	syl6eq 2701	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) = ({0} ∪ (1...𝑁)))
305	304	difeq1d 3760	. . . . . . . 8 ⊢ (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))))
306		difundir 3913	. . . . . . . . . 10 ⊢ (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
307		0lt1 10588	. . . . . . . . . . . . . 14 ⊢ 0 < 1
308		0re 10078	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
309		1re 10077	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
310	308, 309	ltnlei 10196	. . . . . . . . . . . . . 14 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0)
311	307, 310	mpbi 220	. . . . . . . . . . . . 13 ⊢ ¬ 1 ≤ 0
312		elfzle1 12382	. . . . . . . . . . . . 13 ⊢ (0 ∈ (1...(𝑁 − 1)) → 1 ≤ 0)
313	311, 312	mto 188	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ (1...(𝑁 − 1))
314		incom 3838	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑁 − 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 − 1)))
315	314	eqeq1i 2656	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ({0} ∩ (1...(𝑁 − 1))) = ∅)
316		disjsn 4278	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...(𝑁 − 1)))
317		disj3 4054	. . . . . . . . . . . . 13 ⊢ (({0} ∩ (1...(𝑁 − 1))) = ∅ ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
318	315, 316, 317	3bitr3i 290	. . . . . . . . . . . 12 ⊢ (¬ 0 ∈ (1...(𝑁 − 1)) ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
319	313, 318	mpbi 220	. . . . . . . . . . 11 ⊢ {0} = ({0} ∖ (1...(𝑁 − 1)))
320	319	uneq1i 3796	. . . . . . . . . 10 ⊢ ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
321	306, 320	eqtr4i 2676	. . . . . . . . 9 ⊢ (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
322		difundir 3913	. . . . . . . . . . . 12 ⊢ (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
323		difid 3981	. . . . . . . . . . . . 13 ⊢ ((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) = ∅
324	323	uneq1i 3796	. . . . . . . . . . . 12 ⊢ (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
325		uncom 3790	. . . . . . . . . . . . 13 ⊢ (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅)
326		un0 4000	. . . . . . . . . . . . 13 ⊢ (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅) = ({𝑁} ∖ (1...(𝑁 − 1)))
327	325, 326	eqtri 2673	. . . . . . . . . . . 12 ⊢ (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = ({𝑁} ∖ (1...(𝑁 − 1)))
328	322, 324, 327	3eqtri 2677	. . . . . . . . . . 11 ⊢ (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = ({𝑁} ∖ (1...(𝑁 − 1)))
329		nnuz 11761	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
330	1, 329	syl6eleq 2740	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
331	255, 330	eqeltrd 2730	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
332		fzsplit2 12404	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
333	331, 261, 332	syl2anc 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
334	255	oveq1d 6705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
335	1	nnzd 11519	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
336		fzsn 12421	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
337	335, 336	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
338	334, 337	eqtrd 2685	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
339	338	uneq2d 3800	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
340	333, 339	eqtrd 2685	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
341	340	difeq1d 3760	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))))
342	1	nnred 11073	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
343	342	ltm1d 10994	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
344	167	nn0red 11390	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
345	344, 342	ltnled 10222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
346	343, 345	mpbid 222	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
347		elfzle2 12383	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
348	346, 347	nsyl 135	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
349		incom 3838	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 − 1)))
350	349	eqeq1i 2656	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (1...(𝑁 − 1))) = ∅)
351		disjsn 4278	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
352		disj3 4054	. . . . . . . . . . . . 13 ⊢ (({𝑁} ∩ (1...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
353	350, 351, 352	3bitr3i 290	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ (1...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
354	348, 353	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
355	328, 341, 354	3eqtr4a 2711	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = {𝑁})
356	355	uneq2d 3800	. . . . . . . . 9 ⊢ (𝜑 → ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = ({0} ∪ {𝑁}))
357	321, 356	syl5eq 2697	. . . . . . . 8 ⊢ (𝜑 → (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
358	305, 357	eqtrd 2685	. . . . . . 7 ⊢ (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
359	358	eleq2d 2716	. . . . . 6 ⊢ (𝜑 → ((2^nd ‘𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ (2^nd ‘𝑇) ∈ ({0} ∪ {𝑁})))
360		eldif 3617	. . . . . 6 ⊢ ((2^nd ‘𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ ((2^nd ‘𝑇) ∈ (0...𝑁) ∧ ¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))))
361		elun 3786	. . . . . . 7 ⊢ ((2^nd ‘𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2^nd ‘𝑇) ∈ {0} ∨ (2^nd ‘𝑇) ∈ {𝑁}))
362	219	elsn 4225	. . . . . . . 8 ⊢ ((2^nd ‘𝑇) ∈ {0} ↔ (2^nd ‘𝑇) = 0)
363	219	elsn 4225	. . . . . . . 8 ⊢ ((2^nd ‘𝑇) ∈ {𝑁} ↔ (2^nd ‘𝑇) = 𝑁)
364	362, 363	orbi12i 542	. . . . . . 7 ⊢ (((2^nd ‘𝑇) ∈ {0} ∨ (2^nd ‘𝑇) ∈ {𝑁}) ↔ ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁))
365	361, 364	bitri 264	. . . . . 6 ⊢ ((2^nd ‘𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁))
366	359, 360, 365	3bitr3g 302	. . . . 5 ⊢ (𝜑 → (((2^nd ‘𝑇) ∈ (0...𝑁) ∧ ¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) ↔ ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁)))
367	296, 366	bitrd 268	. . . 4 ⊢ (𝜑 → (¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁)))
368	367	biimpa 500	. . 3 ⊢ ((𝜑 ∧ ¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁))
369	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 0) → 𝑁 ∈ ℕ)
370	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 0) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
371	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 0) → 𝑇 ∈ 𝑆)
372		poimirlem22.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
373	372	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) = 0) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
374		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 0) → (2^nd ‘𝑇) = 0)
375	369, 3, 370, 371, 373, 374	poimirlem18 33557	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 0) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
376	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) → 𝑁 ∈ ℕ)
377	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_𝑚 (1...𝑁)))
378	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) → 𝑇 ∈ 𝑆)
379		poimirlem22.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
380	379	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
381		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) → (2^nd ‘𝑇) = 𝑁)
382	376, 3, 377, 378, 380, 381	poimirlem21 33560	. . . 4 ⊢ ((𝜑 ∧ (2^nd ‘𝑇) = 𝑁) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
383	375, 382	jaodan 843	. . 3 ⊢ ((𝜑 ∧ ((2^nd ‘𝑇) = 0 ∨ (2^nd ‘𝑇) = 𝑁)) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
384	368, 383	syldan 486	. 2 ⊢ ((𝜑 ∧ ¬ (2^nd ‘𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
385	293, 384	pm2.61dan 849	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {cab 2637 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 ∃!wreu 2943 ∃*wrmo 2944 {crab 2945 Vcvv 3231 ⦋csb 3566 ∖ cdif 3604 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 ifcif 4119 {csn 4210 {cpr 4212 ⟨cop 4216 class class class wbr 4685 ↦ cmpt 4762 I cid 5052 × cxp 5141 ran crn 5144 ↾ cres 5145 “ cima 5146 ∘ ccom 5147 Fn wfn 5921 ⟶wf 5922 –1-1→wf1 5923 –onto→wfo 5924 –1-1-onto→wf1o 5925 ‘cfv 5926 (class class class)co 6690 ∘_𝑓 cof 6937 1^st c1st 7208 2^nd c2nd 7209 ↑_𝑚 cmap 7899 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 < clt 10112 ≤ cle 10113 − cmin 10304 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ℤ_≥cuz 11725 ...cfz 12364 ..^cfzo 12504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-fac 13101 df-bc 13130 df-hash 13158

This theorem is referenced by: poimirlem27 33566

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