poimirlem16 - Mathbox for Brendan Leahy

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Theorem poimirlem16 35802

Description: Lemma for poimir 35819 establishing the vertices of the simplex of poimirlem17 35803. (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
poimirlem16	⊢ (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})))))

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

**Proof of Theorem poimirlem16**
Step	Hyp	Ref	Expression
1		poimirlem22.2	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		fveq2 6783	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
3	2	breq2d 5087	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
4	3	ifbid 4483	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
5		2fveq3 6788	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
6		2fveq3 6788	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
7	6	imaeq1d 5971	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
8	7	xpeq1d 5619	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
9	6	imaeq1d 5971	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
10	9	xpeq1d 5619	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
11	8, 10	uneq12d 4099	. . . . . . . . 9 ⊢ (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})))
12	5, 11	oveq12d 7302	. . . . . . . 8 ⊢ (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}))))
13	4, 12	csbeq12dv 3842	. . . . . . 7 ⊢ (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	13	mpteq2dv 5177	. . . . . 6 ⊢ (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})))))
15	14	eqeq2d 2750	. . . . 5 ⊢ (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}))))))
16		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}))))}
17	15, 16	elrab2 3628	. . . 4 ⊢ (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}))))))
18	17	simprbi 497	. . 3 ⊢ (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})))))
19	1, 18	syl 17	. 2 ⊢ (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})))))
20		elrabi 3619	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
21	20, 16	eleq2s 2858	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
22	1, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23		xp1st 7872	. . . . . . . . . 10 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25		xp1st 7872	. . . . . . . . 9 ⊢ ((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)))
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
27		elmapfn 8662	. . . . . . . 8 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
29	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
30		1ex 10980	. . . . . . . . . 10 ⊢ 1 ∈ V
31		fnconstg 6671	. . . . . . . . . 10 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1)))
33		c0ex 10978	. . . . . . . . . 10 ⊢ 0 ∈ V
34		fnconstg 6671	. . . . . . . . . 10 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
35	33, 34	ax-mp 5	. . . . . . . . 9 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
36	32, 35	pm3.2i 471	. . . . . . . 8 ⊢ ((((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)))
37		xp2nd 7873	. . . . . . . . . . . . 13 ⊢ ((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)})
38	24, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
39		fvex 6796	. . . . . . . . . . . . 13 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
40		f1oeq1 6713	. . . . . . . . . . . . 13 ⊢ (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)))
41	39, 40	elab 3610	. . . . . . . . . . . 12 ⊢ ((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))
42	38, 41	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
43		dff1o3 6731	. . . . . . . . . . . 12 ⊢ ((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))))
44	43	simprbi 497	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
45	42, 44	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
46		imain 6526	. . . . . . . . . 10 ⊢ (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))))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (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))))
48		elfznn0 13358	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
49		nn0p1nn 12281	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
50	48, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
51	50	nnred 11997	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
52	51	ltp1d 11914	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
53		fzdisj 13292	. . . . . . . . . . . 12 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
55	54	imaeq2d 5972	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
56		ima0 5988	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
57	55, 56	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
58	47, 57	sylan9req 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
59		fnun 6554	. . . . . . . 8 ⊢ ((((((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))))
60	36, 58, 59	sylancr 587	. . . . . . 7 ⊢ ((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))))
61		imaundi 6058	. . . . . . . . 9 ⊢ ((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)))
62		nnuz 12630	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
63	50, 62	eleqtrdi 2850	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
64		peano2uz 12650	. . . . . . . . . . . . 13 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
65	63, 64	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
66		poimir.0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
67	66	nncnd 11998	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
68		npcan1 11409	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
69	67, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
70	69	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
71		elfzuz3 13262	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
72		eluzp1p1 12619	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
73	71, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
74	73	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
75	70, 74	eqeltrrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
76		fzsplit2 13290	. . . . . . . . . . . 12 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
77	65, 75, 76	syl2an2 683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
78	77	imaeq2d 5972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
79		f1ofo 6732	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
80		foima 6702	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
81	42, 79, 80	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
82	81	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
83	78, 82	eqtr3d 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
84	61, 83	eqtr3id 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
85	84	fneq2d 6536	. . . . . . 7 ⊢ ((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)))
86	60, 85	mpbid 231	. . . . . 6 ⊢ ((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))
87		ovexd 7319	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
88		inidm 4153	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
89		eqidd 2740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
90		eqidd 2740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
91	29, 86, 87, 87, 88, 89, 90	offval 7551	. . . . 5 ⊢ ((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}))) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
92		oveq1 7291	. . . . . . . . . 10 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (1 + ((((((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)) = (if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 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}))‘𝑛)))
93	92	eqeq2d 2750	. . . . . . . . 9 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((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)) ↔ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
94		oveq1 7291	. . . . . . . . . 10 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (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}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
95	94	eqeq2d 2750	. . . . . . . . 9 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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}))‘𝑛)) ↔ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
96		1p0e1 12106	. . . . . . . . . . . . . 14 ⊢ (1 + 0) = 1
97	96	eqcomi 2748	. . . . . . . . . . . . 13 ⊢ 1 = (1 + 0)
98		f1ofn 6726	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
99	42, 98	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
100	99	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
101		fzss2 13305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
102	75, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
103		eluzfz1 13272	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑦 + 1)))
104	63, 103	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
105	104	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
106		fnfvima 7118	. . . . . . . . . . . . . . . 16 ⊢ (((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))))
107	100, 102, 105, 106	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
108		fvun1 6868	. . . . . . . . . . . . . . . 16 ⊢ (((((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)))
109	32, 35, 108	mp3an12 1450	. . . . . . . . . . . . . . 15 ⊢ (((((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)))
110	58, 107, 109	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((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)))
111	30	fvconst2 7088	. . . . . . . . . . . . . . 15 ⊢ (((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)
112	107, 111	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
113	110, 112	eqtrd 2779	. . . . . . . . . . . . 13 ⊢ ((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)
114		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
115	66	nnzd 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
116		peano2zm 12372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
117	115, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
118		1z 12359	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℤ
119	117, 118	jctil 520	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
120		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
121	120, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
122		fzaddel 13299	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
123	119, 121, 122	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
124	114, 123	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
125	69	oveq2d 7300	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
126	125	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
127	124, 126	eleqtrd 2842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
128	127	ralrimiva 3104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
129		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
130		peano2z 12370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
131	118, 130	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 + 1) ∈ ℤ
132	115, 131	jctil 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
133		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
134	133, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
135		fzsubel 13301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
136	132, 134, 135	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
137	129, 136	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
138		ax-1cn 10938	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℂ
139	138, 138	pncan3oi 11246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1 + 1) − 1) = 1
140	139	oveq1i 7294	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
141	137, 140	eleqtrdi 2850	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
142	133	zcnd 12436	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
143		elfznn 13294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
144	143	nncnd 11998	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
145		subadd2 11234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
146	138, 145	mp3an2 1448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
147	146	bicomd 222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
148		eqcom 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1))
149		eqcom 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 − 1) = 𝑛 ↔ 𝑛 = (𝑦 − 1))
150	147, 148, 149	3bitr3g 313	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
151	142, 144, 150	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
152	151	ralrimiva 3104	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
153	152	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
154		reu6i 3664	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
155	141, 153, 154	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
156	155	ralrimiva 3104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
157		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
158	157	f1ompt 6994	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((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)))
159	128, 156, 158	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
160		f1osng 6766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
161	66, 30, 160	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
162	66	nnred 11997	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
163	162	ltm1d 11916	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
164	117	zred 12435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
165	164, 162	ltnled 11131	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
166	163, 165	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
167		elfzle2 13269	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
168	166, 167	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
169		disjsn 4648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
170	168, 169	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
171		1re 10984	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
172	171	ltp1i 11888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 < (1 + 1)
173	171, 171	readdcli 10999	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 1) ∈ ℝ
174	171, 173	ltnlei 11105	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
175	172, 174	mpbi 229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ (1 + 1) ≤ 1
176		elfzle1 13268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
177	175, 176	mto 196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
178		disjsn 4648	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
179	177, 178	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
180		f1oun 6744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((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}))
181	179, 180	mpanr2 701	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((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}) = ∅) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
182	159, 161, 170, 181	syl21anc 835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
183	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ V)
184	66, 62	eleqtrdi 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
185	69, 184	eqeltrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
186		uzid 12606	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
187		peano2uz 12650	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
188	117, 186, 187	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
189	69, 188	eqeltrrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
190		fzsplit2 13290	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
191	185, 189, 190	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
192	69	oveq1d 7299	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
193		fzsn 13307	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
194	115, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
195	192, 194	eqtrd 2779	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
196	195	uneq2d 4098	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
197	191, 196	eqtr2d 2780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
198		iftrue 4466	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
199	198	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
200	66, 183, 197, 199	fmptapd 7052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
201		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
202	201	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
203	168, 202	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
204	203	necon2ad 2959	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁))
205	204	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁)
206		ifnefalse 4472	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
207	205, 206	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
208	207	mpteq2dva 5175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
209	208	uneq1d 4097	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
210	200, 209	eqtr3d 2781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
211	191, 196	eqtrd 2779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
212		uzid 12606	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
213		peano2uz 12650	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘1) → (1 + 1) ∈ (ℤ_≥‘1))
214	118, 212, 213	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 1) ∈ (ℤ_≥‘1)
215		fzsplit2 13290	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
216	214, 184, 215	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
217		fzsn 13307	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℤ → (1...1) = {1})
218	118, 217	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...1) = {1}
219	218	uneq1i 4094	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
220	219	equncomi 4090	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
221	216, 220	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
222	210, 211, 221	f1oeq123d 6719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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})))
223	182, 222	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
224		f1oco 6748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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))
225	42, 223, 224	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
226		dff1o3 6731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (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)–onto→(1...𝑁) ∧ Fun ^◡((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))))
227	226	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
228	225, 227	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ^◡((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
229		imain 6526	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ^◡((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)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
230	228, 229	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
231	48	nn0red 12303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
232	231	ltp1d 11914	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
233		fzdisj 13292	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
234	232, 233	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
235	234	imaeq2d 5972	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅))
236		ima0 5988	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) = ∅
237	235, 236	eqtrdi 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
238	230, 237	sylan9req 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅)
239		imassrn 5983	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
240		f1of 6725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁))
241		frn 6616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁) → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁))
242	223, 240, 241	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁))
243	239, 242	sstrid 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
244	243	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
245		elfz1end 13295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
246	66, 245	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
247		eqid 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
248	198, 247, 30	fvmpt 6884	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
249	246, 248	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
250	249	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
251		f1ofn 6726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
252	223, 251	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
253	252	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
254		fzss1 13304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
255	63, 254	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
256	255	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
257		eluzfz2 13273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
258	75, 257	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
259		fnfvima 7118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
260	253, 256, 258, 259	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
261	250, 260	eqeltrrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
262		fnfvima 7118	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁) ∧ 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
263	100, 244, 261, 262	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
264		imaco 6159	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
265	263, 264	eleqtrrdi 2851	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
266		fnconstg 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
267	30, 266	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
268		fnconstg 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
269	33, 268	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
270		fvun2 6869	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∧ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) ∧ (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) → ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘1)))
271	267, 269, 270	mp3an12 1450	. . . . . . . . . . . . . . . 16 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) → ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘1)))
272	238, 265, 271	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘1)))
273	33	fvconst2 7088	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇))‘1) ∈ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 0)
274	265, 273	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 0)
275	272, 274	eqtrd 2779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = 0)
276	275	oveq2d 7300	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1 + ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1))) = (1 + 0))
277	97, 113, 276	3eqtr4a 2805	. . . . . . . . . . . 12 ⊢ ((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 + ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1))))
278		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))
279		fveq2 6783	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((((((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) = ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)))
280	279	oveq2d 7300	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (1 + ((((((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 + ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1))))
281	278, 280	eqeq12d 2755	. . . . . . . . . . . 12 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((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)) ↔ (((((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 + ((((((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}))‘((2^nd ‘(1^st ‘𝑇))‘1)))))
282	277, 281	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((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))))
283	282	imp 407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((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)))
284	283	adantlr 712	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((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)))
285		eldifsn 4721	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘1)))
286		df-ne 2945	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘1) ↔ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1))
287	286	anbi2i 623	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘1)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)))
288	285, 287	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)))
289		fnconstg 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
290	30, 289	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))
291	290, 35	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
292		imain 6526	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
293	45, 292	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
294		fzdisj 13292	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
295	52, 294	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
296	295	imaeq2d 5972	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
297	296, 56	eqtrdi 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
298	293, 297	sylan9req 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
299		fnun 6554	. . . . . . . . . . . . . . . 16 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 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 + 1)...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
300	291, 298, 299	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
301		imaundi 6058	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
302		fzpred 13313	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
303	184, 302	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
304		uncom 4088	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
305	303, 304	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
306	305	difeq1d 4057	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}))
307		difun2 4415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = (((1 + 1)...𝑁) ∖ {1})
308		disj3 4388	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}))
309	179, 308	mpbi 229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})
310	307, 309	eqtr4i 2770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = ((1 + 1)...𝑁)
311	306, 310	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
312	311	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
313		eluzp1p1 12619	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘(1 + 1)))
314	63, 313	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘(1 + 1)))
315		fzsplit2 13290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ_≥‘(1 + 1)) ∧ 𝑁 ∈ (ℤ_≥‘(𝑦 + 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
316	314, 75, 315	syl2an2 683	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
317	312, 316	eqtrd 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
318	317	imaeq2d 5972	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
319		imadif 6525	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {1})))
320	45, 319	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {1})))
321		eluzfz1 13272	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
322	184, 321	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ∈ (1...𝑁))
323		fnsnfv 6856	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑇))‘1)} = ((2^nd ‘(1^st ‘𝑇)) “ {1}))
324	99, 322, 323	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘1)} = ((2^nd ‘(1^st ‘𝑇)) “ {1}))
325	324	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {1}) = {((2^nd ‘(1^st ‘𝑇))‘1)})
326	81, 325	difeq12d 4059	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {1})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
327	320, 326	eqtrd 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
328	327	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
329	318, 328	eqtr3d 2781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
330	301, 329	eqtr3id 2793	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
331	330	fneq2d 6536	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)})))
332	300, 331	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
333		disjdifr 4407	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅
334		fnconstg 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)})
335	30, 334	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)}
336		fvun1 6868	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
337	335, 336	mp3an2 1448	. . . . . . . . . . . . . . . 16 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
338		fnconstg 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)})
339	33, 338	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)}
340		fvun1 6868	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
341	339, 340	mp3an2 1448	. . . . . . . . . . . . . . . 16 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
342	337, 341	eqtr4d 2782	. . . . . . . . . . . . . . 15 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
343	333, 342	mpanr1 700	. . . . . . . . . . . . . 14 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)})) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
344	332, 343	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)})) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
345	288, 344	sylan2br 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1))) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
346	345	anassrs 468	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
347		fzpred 13313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
348	63, 347	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
349	348	imaeq2d 5972	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2^nd ‘(1^st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
350	349	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2^nd ‘(1^st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
351	324	uneq1d 4097	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) = (((2^nd ‘(1^st ‘𝑇)) “ {1}) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))))
352		uncom 4088	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
353		imaundi 6058	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))) = (((2^nd ‘(1^st ‘𝑇)) “ {1}) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
354	351, 352, 353	3eqtr4g 2804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ((2^nd ‘(1^st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
355	354	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ((2^nd ‘(1^st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
356	350, 355	eqtr4d 2782	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) = (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}))
357	356	xpeq1d 5619	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}) × {1}))
358		xpundir 5657	. . . . . . . . . . . . . . . 16 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘1)}) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))
359	357, 358	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1})))
360	359	uneq1d 4097	. . . . . . . . . . . . . 14 ⊢ ((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)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
361		un23 4103	. . . . . . . . . . . . . 14 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))
362	360, 361	eqtrdi 2795	. . . . . . . . . . . . 13 ⊢ ((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)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1})))
363	362	fveq1d 6785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛))
364	363	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {1}))‘𝑛))
365		imaco 6159	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)))
366		df-ima 5603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦))
367		peano2uz 12650	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
368	71, 367	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
369	368	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
370	70, 369	eqeltrrd 2841	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝑦))
371		fzss2 13305	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ_≥‘𝑦) → (1...𝑦) ⊆ (1...𝑁))
372	370, 371	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...𝑁))
373	372	resmptd 5951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
374		fzss2 13305	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
375	71, 374	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
376	375	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
377	168	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
378	376, 377	ssneldd 3925	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦))
379		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦)))
380	379	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦)))
381	378, 380	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦)))
382	381	necon2ad 2959	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛 ≠ 𝑁))
383	382	imp 407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ≠ 𝑁)
384	383, 206	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
385	384	mpteq2dva 5175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
386	373, 385	eqtrd 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
387	386	rneqd 5850	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
388	366, 387	eqtrid 2791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
389		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))
390	389	elrnmpt 5868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
391	390	elv 3439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
392		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
393	392	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ)
394		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦))
395	118	jctl 524	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (1 ∈ ℤ ∧ 𝑦 ∈ ℤ))
396		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ)
397	396, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
398		fzaddel 13299	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
399	395, 397, 398	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
400	394, 399	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
401		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
402	400, 401	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
403	402	rexlimdva 3214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
404		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ)
405	404	zcnd 12436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ)
406		npcan1 11409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗)
407	405, 406	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
408	407	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
409	408	ibir 267	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
410	409	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
411		peano2zm 12372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
412	404, 411	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈ ℤ)
413	412, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
414		fzaddel 13299	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
415	395, 413, 414	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
416	410, 415	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦))
417	405	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈ ℂ)
418	406	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℂ → 𝑗 = ((𝑗 − 1) + 1))
419	417, 418	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 = ((𝑗 − 1) + 1))
420		oveq1 7291	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1))
421	420	rspceeqv 3576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
422	416, 419, 421	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
423	422	ex 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
424	403, 423	impbid 211	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
425	393, 424	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
426	391, 425	syl5bb 283	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
427	426	eqrdv 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1)))
428	388, 427	eqtrd 2779	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1)))
429	428	imaeq2d 5972	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦))) = ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
430	365, 429	eqtrid 2791	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
431	430	xpeq1d 5619	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}))
432		imaundi 6058	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))))
433		imaco 6159	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
434		imaco 6159	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
435	433, 434	uneq12i 4096	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1)))) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
436	432, 435	eqtri 2767	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
437	189	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
438		fzsplit2 13290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
439	73, 437, 438	syl2an2 683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
440	195	uneq2d 4098	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
441	440	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
442	439, 441	eqtrd 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
443		uncom 4088	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))
444	442, 443	eqtrdi 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))))
445	444	imaeq2d 5972	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))))
446	249	sneqd 4574	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1})
447		fnsnfv 6856	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
448	252, 246, 447	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
449	446, 448	eqtr3d 2781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
450	449	imaeq2d 5972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {1}) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
451	324, 450	eqtrd 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘1)} = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
452	451	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2^nd ‘(1^st ‘𝑇))‘1)} = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
453		df-ima 5603	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1)))
454		fzss1 13304	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
455	63, 454	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
456		fzss2 13305	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
457	189, 456	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
458	455, 457	sylan9ssr 3936	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁))
459	458	resmptd 5951	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
460		elfzle2 13269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
461	166, 460	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))
462		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
463	462	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
464	461, 463	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))))
465	464	con2d 134	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → ¬ 𝑛 = 𝑁))
466	465	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁)
467	466	iffalsed 4471	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
468	467	mpteq2dva 5175	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
469	468	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
470	459, 469	eqtrd 2779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
471	470	rneqd 5850	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
472	453, 471	eqtrid 2791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
473		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ)
474	473	zcnd 12436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℂ)
475	474, 406	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) = 𝑗)
476	475	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
477	476	ibir 267	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
478	477	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
479	50	nnzd 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
480	117, 479	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
481	473, 411	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (𝑗 − 1) ∈ ℤ)
482	481, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
483		fzaddel 13299	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
484	480, 482, 483	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
485	478, 484	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → (𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)))
486	474, 418	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 = ((𝑗 − 1) + 1))
487	486	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1))
488	420	rspceeqv 3576	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
489	485, 487, 488	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
490	489	ex 413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
491		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))
492		elfzelz 13265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ)
493	492, 118	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
494		fzaddel 13299	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
495	480, 493, 494	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
496	491, 495	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
497		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
498	496, 497	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
499	498	rexlimdva 3214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
500	490, 499	impbid 211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
501		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))
502	501	elrnmpt 5868	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
503	502	elv 3439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
504	500, 503	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))))
505	504	eqrdv 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
506	69	oveq2d 7300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
507	506	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
508	472, 505, 507	3eqtr2rd 2786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
509	508	imaeq2d 5972	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
510	452, 509	uneq12d 4099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))))
511	436, 445, 510	3eqtr4a 2805	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
512	511	xpeq1d 5619	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}))
513		xpundir 5657	. . . . . . . . . . . . . . . 16 ⊢ (({((2^nd ‘(1^st ‘𝑇))‘1)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
514	512, 513	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
515	431, 514	uneq12d 4099	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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})) = ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
516		unass 4101	. . . . . . . . . . . . . . 15 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
517		un23 4103	. . . . . . . . . . . . . . 15 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))
518	516, 517	eqtr3i 2769	. . . . . . . . . . . . . 14 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))
519	515, 518	eqtrdi 2795	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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})) = (((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0})))
520	519	fveq1d 6785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((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) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
521	520	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → ((((((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) = ((((((2^nd ‘(1^st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘1)} × {0}))‘𝑛))
522	346, 364, 521	3eqtr4d 2789	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((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))
523		snssi 4742	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ)
524	138, 523	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {1} ⊆ ℂ
525		0cn 10976	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
526		snssi 4742	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
527	525, 526	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℂ
528	524, 527	unssi 4120	. . . . . . . . . . . . 13 ⊢ ({1} ∪ {0}) ⊆ ℂ
529	30	fconst 6669	. . . . . . . . . . . . . . . . 17 ⊢ ((((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)))) “ (1...𝑦))⟶{1}
530	33	fconst 6669	. . . . . . . . . . . . . . . . 17 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}
531	529, 530	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ (((((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)))) “ (1...𝑦))⟶{1} ∧ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0})
532		fun 6645	. . . . . . . . . . . . . . . 16 ⊢ (((((((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)))) “ (1...𝑦))⟶{1} ∧ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}) ∧ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅) → (((((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})):((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
533	531, 238, 532	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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})):((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
534		imaundi 6058	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
535		fzsplit2 13290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
536	63, 370, 535	syl2an2 683	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
537	536	imaeq2d 5972	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
538		f1ofo 6732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (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)–onto→(1...𝑁))
539		foima 6702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
540	225, 538, 539	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
541	540	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
542	537, 541	eqtr3d 2781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
543	534, 542	eqtr3id 2793	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
544	543	feq2d 6595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((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})):((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {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})):(1...𝑁)⟶({1} ∪ {0})))
545	533, 544	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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})):(1...𝑁)⟶({1} ∪ {0}))
546	545	ffvelrnda 6970	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((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} ∪ {0}))
547	528, 546	sselid 3920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((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) ∈ ℂ)
548	547	addid2d 11185	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (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}))‘𝑛)) = ((((((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))
549	548	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (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}))‘𝑛)) = ((((((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))
550	522, 549	eqtr4d 2782	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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}))‘𝑛)))
551	93, 95, 284, 550	ifbothda 4498	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
552	551	oveq2d 7300	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
553		elmapi 8646	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
554	26, 553	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
555	554	ffvelrnda 6970	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
556		elfzonn0 13441	. . . . . . . . . . 11 ⊢ (((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
557	555, 556	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
558	557	nn0cnd 12304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
559	558	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
560	138, 525	ifcli 4507	. . . . . . . . 9 ⊢ if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) ∈ ℂ
561	560	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) ∈ ℂ)
562	559, 561, 547	addassd 11006	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
563	552, 562	eqtr4d 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
564	563	mpteq2dva 5175	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
565	91, 564	eqtrd 2779	. . . 4 ⊢ ((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}))) = (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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
566		poimirlem18.4	. . . . . . . . . 10 ⊢ (𝜑 → (2^nd ‘𝑇) = 0)
567	566	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) = 0)
568		elfzle1 13268	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
569	568	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
570	567, 569	eqbrtrd 5097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) ≤ 𝑦)
571		0re 10986	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
572	566, 571	eqeltrdi 2848	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
573		lenlt 11062	. . . . . . . . 9 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
574	572, 231, 573	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
575	570, 574	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2^nd ‘𝑇))
576	575	iffalsed 4471	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
577	576	csbeq1d 3837	. . . . 5 ⊢ ((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}))))
578		ovex 7317	. . . . . 6 ⊢ (𝑦 + 1) ∈ V
579		oveq2 7292	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
580	579	imaeq2d 5972	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
581	580	xpeq1d 5619	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
582		oveq1 7291	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
583	582	oveq1d 7299	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
584	583	imaeq2d 5972	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
585	584	xpeq1d 5619	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
586	581, 585	uneq12d 4099	. . . . . . 7 ⊢ (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})))
587	586	oveq2d 7300	. . . . . 6 ⊢ (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}))))
588	578, 587	csbie 3869	. . . . 5 ⊢ ⦋(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})))
589	577, 588	eqtrdi 2795	. . . 4 ⊢ ((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}))))
590		ovexd 7319	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ V)
591		fvexd 6798	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((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) ∈ V)
592		eqidd 2740	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))))
593	545	ffnd 6610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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})) Fn (1...𝑁))
594		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(2^nd ‘(1^st ‘𝑇))
595		nfmpt1 5183	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
596	594, 595	nfco 5777	. . . . . . . . . 10 ⊢ Ⅎ𝑛((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
597		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1...𝑦)
598	596, 597	nfima 5980	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
599		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑛{1}
600	598, 599	nfxp 5623	. . . . . . . 8 ⊢ Ⅎ𝑛((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1})
601		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑦 + 1)...𝑁)
602	596, 601	nfima 5980	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
603		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑛{0}
604	602, 603	nfxp 5623	. . . . . . . 8 ⊢ Ⅎ𝑛((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})
605	600, 604	nfun 4100	. . . . . . 7 ⊢ Ⅎ𝑛(((((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}))
606	605	dffn5f 6849	. . . . . 6 ⊢ ((((((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})) Fn (1...𝑁) ↔ (((((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) ↦ ((((((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)))
607	593, 606	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((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) ↦ ((((((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)))
608	87, 590, 591, 592, 607	offval2 7562	. . . 4 ⊢ ((𝜑 ∧ 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)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
609	565, 589, 608	3eqtr4rd 2790	. . 3 ⊢ ((𝜑 ∧ 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}))) = ⦋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}))))
610	609	mpteq2dva 5175	. 2 ⊢ (𝜑 → (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})))) = (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})))))
611	19, 610	eqtr4d 2782	1 ⊢ (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})))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2107 {cab 2716 ≠ wne 2944 ∀wral 3065 ∃wrex 3066 ∃!wreu 3067 {crab 3069 Vcvv 3433 ⦋csb 3833 ∖ cdif 3885 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ∅c0 4257 ifcif 4460 {csn 4562 ⟨cop 4568 class class class wbr 5075 ↦ cmpt 5158 × cxp 5588 ^◡ccnv 5589 ran crn 5591 ↾ cres 5592 “ cima 5593 ∘ ccom 5594 Fun wfun 6431 Fn wfn 6432 ⟶wf 6433 –onto→wfo 6435 –1-1-onto→wf1o 6436 ‘cfv 6437 (class class class)co 7284 ∘_f cof 7540 1^st c1st 7838 2^nd c2nd 7839 ↑_m cmap 8624 ℂcc 10878 ℝcr 10879 0cc0 10880 1c1 10881 + caddc 10883 < clt 11018 ≤ cle 11019 − cmin 11214 ℕcn 11982 ℕ₀cn0 12242 ℤcz 12328 ℤ_≥cuz 12591 ...cfz 13248 ..^cfzo 13391

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 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-of 7542 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-er 8507 df-map 8626 df-en 8743 df-dom 8744 df-sdom 8745 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-n0 12243 df-z 12329 df-uz 12592 df-fz 13249 df-fzo 13392

This theorem is referenced by: poimirlem17 35803

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