poimirlem17 - Mathbox for Brendan Leahy

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Theorem poimirlem17 36494

Description: Lemma for poimir 36510 establishing existence for poimirlem18 36495. (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)
poimirlem18.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
poimirlem18.4	⊢ (𝜑 → (2^nd ‘𝑇) = 0)

**Assertion**
Ref	Expression
poimirlem17	⊢ (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 poimirlem17**
Step	Hyp	Ref	Expression
1		poimir.0	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		poimirlem22.s	. . . . 5 ⊢ 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}))))}
3		poimirlem22.1	. . . . 5 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
4		poimirlem22.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		poimirlem18.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
6		poimirlem18.4	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑇) = 0)
7	1, 2, 3, 4, 5, 6	poimirlem16 36493	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
8		elfznn0 13591	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
9	8	nn0red 12530	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
10	9	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
11	1	nnzd 12582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12		peano2zm 12602	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
13	11, 12	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
14	13	zred 12663	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
15	14	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
16	1	nnred 12224	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
18		elfzle2 13502	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
19	18	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
20	16	ltm1d 12143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
21	20	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
22	10, 15, 17, 19, 21	lelttrd 11369	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
23	22	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
24		fveq2 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (2^nd ‘𝑡) = (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
25		opex 5464	. . . . . . . . . . . . . . . 16 ⊢ ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V
26		op2ndg 7985	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
27	25, 1, 26	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
28	24, 27	sylan9eqr 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘𝑡) = 𝑁)
29	28	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑡) = 𝑁)
30	23, 29	breqtrrd 5176	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2^nd ‘𝑡))
31	30	iftrued 4536	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = 𝑦)
32	31	csbeq1d 3897	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 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 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
33		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		oveq2 7414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
35	34	imaeq2d 6058	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)))
36	35	xpeq1d 5705	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}))
37		oveq1 7413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
38	37	oveq1d 7421	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
39	38	imaeq2d 6058	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)))
40	39	xpeq1d 5705	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))
41	36, 40	uneq12d 4164	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
42	41	oveq2d 7422	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → ((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...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))))
43	33, 42	csbie 3929	. . . . . . . . . . . 12 ⊢ ⦋𝑦 / 𝑗⦌((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...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
44		2fveq3 6894	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)))
45		op1stg 7984	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
46	25, 1, 45	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
47	46	fveq2d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (1^st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
48		ovex 7439	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
49	48	mptex 7222	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ V
50		fvex 6902	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
51	48	mptex 7222	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∈ V
52	50, 51	coex 7918	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ V
53	49, 52	op1st 7980	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
54	47, 53	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))))
55	44, 54	sylan9eqr 2795	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1^st ‘(1^st ‘𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))))
56		fveq2 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1^st ‘𝑡) = (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
57	56, 46	sylan9eqr 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1^st ‘𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
58	57	fveq2d 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
59	49, 52	op2nd 7981	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
60	58, 59	eqtrdi 2789	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
61	60	imaeq1d 6057	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
62	61	xpeq1d 5705	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}))
63	60	imaeq1d 6057	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
64	63	xpeq1d 5705	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))
65	62, 64	uneq12d 4164	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))
66	55, 65	oveq12d 7424	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
67	43, 66	eqtrid 2785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
68	67	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
69	32, 68	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 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}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
70	69	mpteq2dva 5248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (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)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
71	70	eqeq2d 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
72	71	ex 414	. . . . . 6 ⊢ (𝜑 → (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
73	72	alrimiv 1931	. . . . 5 ⊢ (𝜑 → ∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
74		oveq2 7414	. . . . . . . . . . 11 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
75	74	eleq1d 2819	. . . . . . . . . 10 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
76		oveq2 7414	. . . . . . . . . . 11 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
77	76	eleq1d 2819	. . . . . . . . . 10 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
78		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)))
79	78	oveq1d 7421	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
80	79	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
81		elrabi 3677	. . . . . . . . . . . . . . . . . . 19 ⊢ (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)))
82	81, 2	eleq2s 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83		xp1st 8004	. . . . . . . . . . . . . . . . . 18 ⊢ (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)}))
84	4, 82, 83	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85		xp1st 8004	. . . . . . . . . . . . . . . . 17 ⊢ ((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)))
86		elmapi 8840	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
87	84, 85, 86	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
88	4, 82	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89		xp2nd 8005	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)})
90	88, 83, 89	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
91		f1oeq1 6819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)))
92	50, 91	elab 3668	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))
93	90, 92	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
94		f1of 6831	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
96		nnuz 12862	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ_≥‘1)
97	1, 96	eleqtrdi 2844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
98		eluzfz1 13505	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
99	97, 98	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ (1...𝑁))
100	95, 99	ffvelcdmd 7085	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁))
101	87, 100	ffvelcdmd 7085	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾))
102		elfzonn0 13674	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀)
103		peano2nn0 12509	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀)
104	101, 102, 103	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀)
105		elfzo0 13670	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾))
106	101, 105	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾))
107	106	simp2d 1144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ ℕ)
108	104	nn0red 12530	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℝ)
109	107	nnred 12224	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ℝ)
110		elfzolt2 13638	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾)
111	101, 110	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾)
112	101, 102	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀)
113	112	nn0zd 12581	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℤ)
114	107	nnzd 12582	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℤ)
115		zltp1le 12609	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾 ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾))
116	113, 114, 115	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾 ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾))
117	111, 116	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾)
118		fvex 6902	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ V
119		eleq1 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (𝑛 ∈ (1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)))
120	119	anbi2d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁))))
121		fveq2 6889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (𝑝‘𝑛) = (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)))
122	121	neeq1d 3001	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾))
123	122	rexbidv 3179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾))
124	120, 123	imbi12d 345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) ↔ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)))
125	118, 124, 5	vtocl 3550	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)
126	100, 125	mpdan 686	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)
127		fveq1 6888	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)))
128	87	ffnd 6716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
129	128	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
130		1ex 11207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
131		fnconstg 6777	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
132	130, 131	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1)))
133		c0ex 11205	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ V
134		fnconstg 6777	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
135	133, 134	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
136	132, 135	pm3.2i 472	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
137		dff1o3 6837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑇))))
138	137	simprbi 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
139		imain 6631	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
140	93, 138, 139	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
141		nn0p1nn 12508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
142	8, 141	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
143	142	nnred 12224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
144	143	ltp1d 12141	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
145		fzdisj 13525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
146	145	imaeq2d 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
147		ima0 6074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
148	146, 147	eqtrdi 2789	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
149	144, 148	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
150	140, 149	sylan9req 2794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
151		fnun 6661	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
152	136, 150, 151	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
153		imaundi 6147	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
154	142	peano2nnd 12226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
155	154, 96	eleqtrdi 2844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
156	1	nncnd 12225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑁 ∈ ℂ)
157		npcan1 11636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
158	156, 157	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
159	158	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
160		elfzuz3 13495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
161		eluzp1p1 12847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
162	160, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
163	162	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
164	159, 163	eqeltrrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
165		fzsplit2 13523	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
166	155, 164, 165	syl2an2 685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
167	166	imaeq2d 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
168		f1ofo 6838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
169		foima 6808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
170	93, 168, 169	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
171	170	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
172	167, 171	eqtr3d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
173	153, 172	eqtr3id 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
174	173	fneq2d 6641	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
175	152, 174	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
176	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
177		inidm 4218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
178		eqidd 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)))
179		f1ofn 6832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
180	93, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
181	180	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
182		fzss2 13538	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
183	164, 182	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
184	142, 96	eleqtrdi 2844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
185		eluzfz1 13505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑦 + 1)))
186	184, 185	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
187	186	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
188		fnfvima 7232	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
189	181, 183, 187, 188	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
190		fvun1 6980	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
191	132, 135, 190	mp3an12 1452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
192	150, 189, 191	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
193	130	fvconst2 7202	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
194	189, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
195	192, 194	eqtrd 2773	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
196	195	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
197	129, 175, 176, 176, 177, 178, 196	ofval 7678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
198	100, 197	mpidan 688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
199	127, 198	sylan9eqr 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
200	199	adantllr 718	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
201		fveq2 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
202	201	breq2d 5160	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
203	202	ifbid 4551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
204		2fveq3 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
205		2fveq3 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
206	205	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
207	206	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
208	205	imaeq1d 6057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
209	208	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
210	207, 209	uneq12d 4164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (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})))
211	204, 210	oveq12d 7424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (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}))))
212	203, 211	csbeq12dv 3902	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (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}))))
213	212	mpteq2dv 5250	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (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})))))
214	213	eqeq2d 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (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}))))))
215	214, 2	elrab2 3686	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (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}))))))
216	215	simprbi 498	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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})))))
217	4, 216	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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})))))
218	217	rneqd 5936	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ran 𝐹 = ran (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})))))
219	218	eleq2d 2820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ 𝑝 ∈ ran (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}))))))
220		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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}))))
221		ovex 7439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
222	221	csbex 5311	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⦋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}))) ∈ V
223	220, 222	elrnmpti 5958	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ ran (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))𝑝 = ⦋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}))))
224	219, 223	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ⦋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})))))
225	6	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) = 0)
226		elfzle1 13501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
227	226	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
228	225, 227	eqbrtrd 5170	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) ≤ 𝑦)
229		0re 11213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℝ
230	6, 229	eqeltrdi 2842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
231		lenlt 11289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
232	230, 9, 231	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
233	228, 232	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2^nd ‘𝑇))
234	233	iffalsed 4539	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
235	234	csbeq1d 3897	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 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 + 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}))))
236		ovex 7439	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 + 1) ∈ V
237		oveq2 7414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
238	237	imaeq2d 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
239	238	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
240		oveq1 7413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
241	240	oveq1d 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
242	241	imaeq2d 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
243	242	xpeq1d 5705	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
244	239, 243	uneq12d 4164	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
245	244	oveq2d 7422	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = (𝑦 + 1) → ((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...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
246	236, 245	csbie 3929	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⦋(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}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
247	235, 246	eqtrdi 2789	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 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}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
248	247	eqeq2d 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑝 = ⦋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}))) ↔ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
249	248	rexbidva 3177	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ⦋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))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
250	224, 249	bitrd 279	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
251	250	biimpa 478	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
252	200, 251	r19.29a 3163	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
253		eqtr3 2759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∧ 𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾)
254	253	ex 414	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾))
255	252, 254	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → (𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾))
256	255	necon3d 2962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)))
257	256	rexlimdva 3156	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)))
258	126, 257	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
259	108, 109, 117, 258	leneltd 11365	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) < 𝐾)
260		elfzo0 13670	. . . . . . . . . . . . . 14 ⊢ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾) ↔ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) < 𝐾))
261	104, 107, 259, 260	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾))
262	261	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾))
263	80, 262	eqeltrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
264	263	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
265	87	ffvelcdmda 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
266		elfzonn0 13674	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
267	265, 266	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
268	267	nn0cnd 12531	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
269	268	addridd 11411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
270	269, 265	eqeltrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
271	270	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
272	75, 77, 264, 271	ifbothda 4566	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾))
273	272	fmpttd 7112	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
274		ovex 7439	. . . . . . . . 9 ⊢ (0..^𝐾) ∈ V
275	274, 48	elmap 8862	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
276	273, 275	sylibr 233	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
277		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
278		1z 12589	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
279	13, 278	jctil 521	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
280		elfzelz 13498	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
281	280, 278	jctir 522	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
282		fzaddel 13532	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
283	279, 281, 282	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
284	277, 283	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
285	158	oveq2d 7422	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
286	285	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
287	284, 286	eleqtrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
288	287	ralrimiva 3147	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
289		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
290		peano2z 12600	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
291	278, 290	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 1) ∈ ℤ
292	11, 291	jctil 521	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
293		elfzelz 13498	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
294	293, 278	jctir 522	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
295		fzsubel 13534	. . . . . . . . . . . . . . . . 17 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
296	292, 294, 295	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
297	289, 296	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
298		ax-1cn 11165	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
299	298, 298	pncan3oi 11473	. . . . . . . . . . . . . . . 16 ⊢ ((1 + 1) − 1) = 1
300	299	oveq1i 7416	. . . . . . . . . . . . . . 15 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
301	297, 300	eleqtrdi 2844	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
302	293	zcnd 12664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
303		elfznn 13527	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
304	303	nncnd 12225	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
305		subadd2 11461	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
306	298, 305	mp3an2 1450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
307	306	bicomd 222	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
308		eqcom 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑛 + 1) ↔ (𝑛 + 1) = 𝑦)
309		eqcom 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑛)
310	307, 308, 309	3bitr4g 314	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
311	302, 304, 310	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
312	311	ralrimiva 3147	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
313	312	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
314		reu6i 3724	. . . . . . . . . . . . . 14 ⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
315	301, 313, 314	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
316	315	ralrimiva 3147	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
317		eqid 2733	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
318	317	f1ompt 7108	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)))
319	288, 316, 318	sylanbrc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
320		f1osng 6872	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
321	1, 130, 320	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
322	14, 16	ltnled 11358	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
323	20, 322	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
324		elfzle2 13502	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
325	323, 324	nsyl 140	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
326		disjsn 4715	. . . . . . . . . . . 12 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
327	325, 326	sylibr 233	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
328		1re 11211	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
329	328	ltp1i 12115	. . . . . . . . . . . . . . 15 ⊢ 1 < (1 + 1)
330	291	zrei 12561	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) ∈ ℝ
331	328, 330	ltnlei 11332	. . . . . . . . . . . . . . 15 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
332	329, 331	mpbi 229	. . . . . . . . . . . . . 14 ⊢ ¬ (1 + 1) ≤ 1
333		elfzle1 13501	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
334	332, 333	mto 196	. . . . . . . . . . . . 13 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
335		disjsn 4715	. . . . . . . . . . . . 13 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
336	334, 335	mpbir 230	. . . . . . . . . . . 12 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
337	336	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (((1 + 1)...𝑁) ∩ {1}) = ∅)
338		f1oun 6850	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ∧ {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1}) ∧ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ∧ (((1 + 1)...𝑁) ∩ {1}) = ∅)) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
339	319, 321, 327, 337, 338	syl22anc 838	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
340	130	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ V)
341	158, 97	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
342		uzid 12834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
343		peano2uz 12882	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
344	13, 342, 343	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
345	158, 344	eqeltrrd 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
346		fzsplit2 13523	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
347	341, 345, 346	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
348	158	oveq1d 7421	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
349		fzsn 13540	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
350	11, 349	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
351	348, 350	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
352	351	uneq2d 4163	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
353	347, 352	eqtr2d 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
354		iftrue 4534	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
355	354	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
356	1, 340, 353, 355	fmptapd 7166	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
357		eleq1 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
358	357	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
359	325, 358	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
360	359	necon2ad 2956	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁))
361	360	imp 408	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁)
362		ifnefalse 4540	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
363	361, 362	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
364	363	mpteq2dva 5248	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
365	364	uneq1d 4162	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
366	356, 365	eqtr3d 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
367	347, 352	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
368		uzid 12834	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
369		peano2uz 12882	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (ℤ_≥‘1) → (1 + 1) ∈ (ℤ_≥‘1))
370	278, 368, 369	mp2b 10	. . . . . . . . . . . . 13 ⊢ (1 + 1) ∈ (ℤ_≥‘1)
371		fzsplit2 13523	. . . . . . . . . . . . 13 ⊢ (((1 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
372	370, 97, 371	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
373		fzsn 13540	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (1...1) = {1})
374	278, 373	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (1...1) = {1}
375	374	uneq1i 4159	. . . . . . . . . . . . 13 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
376	375	equncomi 4155	. . . . . . . . . . . 12 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
377	372, 376	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
378	366, 367, 377	f1oeq123d 6825	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1})))
379	339, 378	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
380		f1oco 6854	. . . . . . . . 9 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
381	93, 379, 380	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
382		f1oeq1 6819	. . . . . . . . 9 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)))
383	52, 382	elab 3668	. . . . . . . 8 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
384	381, 383	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
385	276, 384	opelxpd 5714	. . . . . 6 ⊢ (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
386	1	nnnn0d 12529	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
387		nn0fz0 13596	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
388	386, 387	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
389	385, 388	opelxpd 5714	. . . . 5 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
390		elrab3t 3682	. . . . 5 ⊢ ((∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))) ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
391	73, 389, 390	syl2anc 585	. . . 4 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
392	7, 391	mpbird 257	. . 3 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))})
393	392, 2	eleqtrrdi 2845	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆)
394		fveqeq2 6898	. . . . . 6 ⊢ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → ((2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁 ↔ (2^nd ‘𝑇) = 𝑁))
395	27, 394	syl5ibcom 244	. . . . 5 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2^nd ‘𝑇) = 𝑁))
396	1	nnne0d 12259	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
397		neeq1 3004	. . . . . 6 ⊢ ((2^nd ‘𝑇) = 𝑁 → ((2^nd ‘𝑇) ≠ 0 ↔ 𝑁 ≠ 0))
398	396, 397	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → ((2^nd ‘𝑇) = 𝑁 → (2^nd ‘𝑇) ≠ 0))
399	395, 398	syld 47	. . . 4 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2^nd ‘𝑇) ≠ 0))
400	399	necon2d 2964	. . 3 ⊢ (𝜑 → ((2^nd ‘𝑇) = 0 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
401	6, 400	mpd 15	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇)
402		neeq1 3004	. . 3 ⊢ (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝑧 ≠ 𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
403	402	rspcev 3613	. 2 ⊢ ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇) → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
404	393, 401, 403	syl2anc 585	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 {crab 3433 Vcvv 3475 ⦋csb 3893 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 {csn 4628 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 ran crn 5677 “ cima 5679 ∘ ccom 5680 Fun wfun 6535 Fn wfn 6536 ⟶wf 6537 –onto→wfo 6539 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∘_f cof 7665 1^st c1st 7970 2^nd c2nd 7971 ↑_m cmap 8817 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 < clt 11245 ≤ cle 11246 − cmin 11441 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ℤ_≥cuz 12819 ...cfz 13481 ..^cfzo 13624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625

This theorem is referenced by: poimirlem18 36495

W3C validator