poimirlem16 - Mathbox for Brendan Leahy

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Theorem poimirlem16 38073

Description: Lemma for poimir 38090 establishing the vertices of the simplex of poimirlem17 38074. (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 6852	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
3	2	breq2d 5102	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
4	3	ifbid 4494	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
5		2fveq3 6857	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
6		2fveq3 6857	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
7	6	imaeq1d 6034	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
8	7	xpeq1d 5665	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
9	6	imaeq1d 6034	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
10	9	xpeq1d 5665	. . . . . . . . . 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 4113	. . . . . . . . 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 7399	. . . . . . . 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 3852	. . . . . . 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 5184	. . . . . 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 2763	. . . . 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 3644	. . . 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 500	. . 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 3637	. . . . . . . . . . . 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 2870	. . . . . . . . . . 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 7987	. . . . . . . . . 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 7987	. . . . . . . . 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 8831	. . . . . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
30		1ex 11162	. . . . . . . . . 10 ⊢ 1 ∈ V
31		fnconstg 6737	. . . . . . . . . 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 11159	. . . . . . . . . 10 ⊢ 0 ∈ V
34		fnconstg 6737	. . . . . . . . . 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 473	. . . . . . . 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 7988	. . . . . . . . . . . . 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 6865	. . . . . . . . . . . . 13 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
40		f1oeq1 6779	. . . . . . . . . . . . 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 3629	. . . . . . . . . . . 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 220	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
43		dff1o3 6798	. . . . . . . . . . . 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 500	. . . . . . . . . . 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 6591	. . . . . . . . . 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 13611	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
49		nn0p1nn 12506	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
50	48, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
51	50	nnred 12211	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
52	51	ltp1d 12108	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
53		fzdisj 13542	. . . . . . . . . . . 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 6035	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
56		ima0 6052	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
57	55, 56	eqtrdi 2803	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
58	47, 57	sylan9req 2808	. . . . . . . 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 6620	. . . . . . . 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 595	. . . . . . 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 6120	. . . . . . . . 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 12864	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
63	50, 62	eleqtrdi 2862	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
64		peano2uz 12888	. . . . . . . . . . . . 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 12212	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
68		npcan1 11598	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
69	67, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
70	69	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
71		elfzuz3 13512	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
72		eluzp1p1 12853	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
75	70, 74	eqeltrrd 2853	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
76		fzsplit2 13540	. . . . . . . . . . . 12 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
77	65, 75, 76	syl2an2 694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
78	77	imaeq2d 6035	. . . . . . . . . 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 6799	. . . . . . . . . . . 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 6768	. . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
83	78, 82	eqtr3d 2789	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
84	61, 83	eqtr3id 2801	. . . . . . . 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 6600	. . . . . . 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 234	. . . . . 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 7416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
88		inidm 4169	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
89		eqidd 2753	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
90		eqidd 2753	. . . . . 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 7654	. . . . 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 7388	. . . . . . . . . 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 2763	. . . . . . . . 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 7388	. . . . . . . . . 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 2763	. . . . . . . . 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 12326	. . . . . . . . . . . . . 14 ⊢ (1 + 0) = 1
97	96	eqcomi 2761	. . . . . . . . . . . . 13 ⊢ 1 = (1 + 0)
98		f1ofn 6792	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
101		fzss2 13555	. . . . . . . . . . . . . . . . 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 13522	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
106		fnfvima 7202	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
108		fvun1 6943	. . . . . . . . . . . . . . . 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 1462	. . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . 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 7173	. . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
115	66	nnzd 12580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
116		peano2zm 12600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
117	115, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
118		1z 12587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℤ
119	117, 118	jctil 526	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
120		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
121	120, 118	jctir 527	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
122		fzaddel 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
124	114, 123	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
125	69	oveq2d 7397	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
126	125	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
127	124, 126	eleqtrd 2854	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
128	127	ralrimiva 3144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
129		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
130		peano2z 12598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
131	118, 130	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 + 1) ∈ ℤ
132	115, 131	jctil 526	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
133		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
134	133, 118	jctir 527	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
135		fzsubel 13551	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
137	129, 136	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
138		ax-1cn 11117	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℂ
139	138, 138	pncan3oi 11432	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1 + 1) − 1) = 1
140	139	oveq1i 7391	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
141	137, 140	eleqtrdi 2862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
142	133	zcnd 12664	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
143		elfznn 13544	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
144	143	nncnd 12212	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
145		subadd2 11420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
146	138, 145	mp3an2 1460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
147	146	bicomd 225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
148		eqcom 2759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1))
149		eqcom 2759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 − 1) = 𝑛 ↔ 𝑛 = (𝑦 − 1))
150	147, 148, 149	3bitr3g 315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
151	142, 144, 150	syl2an 604	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
152	151	ralrimiva 3144	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
153	152	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
154		reu6i 3681	. . . . . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
156	155	ralrimiva 3144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
157		eqid 2752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
158	157	f1ompt 7077	. . . . . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
160		f1osng 6834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
161	66, 30, 160	sylancl 594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
162	66	nnred 12211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
163	162	ltm1d 12110	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
164	117	zred 12663	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
165	164, 162	ltnled 11316	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
166	163, 165	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
167		elfzle2 13519	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
168	166, 167	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
169		disjsn 4660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
170	168, 169	sylibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
171		1re 11167	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
172	171	ltp1i 12082	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 < (1 + 1)
173	171, 171	readdcli 11183	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 1) ∈ ℝ
174	171, 173	ltnlei 11290	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
175	172, 174	mpbi 232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ (1 + 1) ≤ 1
176		elfzle1 13518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
177	175, 176	mto 199	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
178		disjsn 4660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
179	177, 178	mpbir 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
180		f1oun 6811	. . . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . . 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 846	. . . . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
185	69, 184	eqeltrd 2852	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
186		uzid 12840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
187		peano2uz 12888	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
190		fzsplit2 13540	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
191	185, 189, 190	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
192	69	oveq1d 7396	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
193		fzsn 13557	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
194	115, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
195	192, 194	eqtrd 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
196	195	uneq2d 4112	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
197	191, 196	eqtr2d 2788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
198		iftrue 4476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
199	198	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
200	66, 183, 197, 199	fmptapd 7140	. . . . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
202	201	notbid 320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
203	168, 202	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
204	203	necon2ad 2962	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁))
205	204	imp 409	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁)
206		ifnefalse 4482	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5183	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
209	208	uneq1d 4111	. . . . . . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
211	191, 196	eqtrd 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
212		uzid 12840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
213		peano2uz 12888	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ_≥‘1) → (1 + 1) ∈ (ℤ_≥‘1))
214	118, 212, 213	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 1) ∈ (ℤ_≥‘1)
215		fzsplit2 13540	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
216	214, 184, 215	sylancr 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
217		fzsn 13557	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℤ → (1...1) = {1})
218	118, 217	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...1) = {1}
219	218	uneq1i 4108	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
220	219	equncomi 4104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
221	216, 220	eqtrdi 2803	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
222	210, 211, 221	f1oeq123d 6785	. . . . . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
224		f1oco 6815	. . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . 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 6798	. . . . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . . 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 6591	. . . . . . . . . . . . . . . . . 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 12529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
232	231	ltp1d 12108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
233		fzdisj 13542	. . . . . . . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . 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 6052	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) = ∅
237	235, 236	eqtrdi 2803	. . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . . . . . . . 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 6791	. . . . . . . . . . . . . . . . . . . . 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 6684	. . . . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
244	243	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
245		elfz1end 13545	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
246	66, 245	sylib 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
247		eqid 2752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
248	198, 247, 30	fvmpt 6960	. . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
251		f1ofn 6792	. . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
254		fzss1 13554	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
257		eluzfz2 13523	. . . . . . . . . . . . . . . . . . . . 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 7202	. . . . . . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
262		fnfvima 7202	. . . . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . . . 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 6223	. . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . 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 6944	. . . . . . . . . . . . . . . . 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 1462	. . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . 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 7173	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . 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 2813	. . . . . . . . . . . 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 6852	. . . . . . . . . . . . 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 6852	. . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 249	. . . . . . . . . . 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 409	. . . . . . . . . 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 723	. . . . . . . . 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 4736	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘1)))
286		df-ne 2948	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘1) ↔ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1))
287	286	anbi2i 631	. . . . . . . . . . . . . 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 277	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)))
289		fnconstg 6737	. . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . 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 6591	. . . . . . . . . . . . . . . . . 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 13542	. . . . . . . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
298	293, 297	sylan9req 2808	. . . . . . . . . . . . . . . 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 6620	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 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 6120	. . . . . . . . . . . . . . . . 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 13563	. . . . . . . . . . . . . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
305	303, 304	eqtrdi 2803	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
306	305	difeq1d 4070	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}))
307		difun2 4425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = (((1 + 1)...𝑁) ∖ {1})
308		disj3 4398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}))
309	179, 308	mpbi 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})
310	307, 309	eqtr4i 2778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = ((1 + 1)...𝑁)
311	306, 310	eqtrdi 2803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
312	311	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
313		eluzp1p1 12853	. . . . . . . . . . . . . . . . . . . . . 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 13540	. . . . . . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
317	312, 316	eqtrd 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
318	317	imaeq2d 6035	. . . . . . . . . . . . . . . . . 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 6590	. . . . . . . . . . . . . . . . . . . . 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 13522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
322	184, 321	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ∈ (1...𝑁))
323		fnsnfv 6931	. . . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘1)} = ((2^nd ‘(1^st ‘𝑇)) “ {1}))
325	324	eqcomd 2758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {1}) = {((2^nd ‘(1^st ‘𝑇))‘1)})
326	81, 325	difeq12d 4072	. . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}))
328	327	adantr 483	. . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . 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 6600	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 4417	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘1)}) ∩ {((2^nd ‘(1^st ‘𝑇))‘1)}) = ∅
334		fnconstg 6737	. . . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . 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 1460	. . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . 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 1460	. . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 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 470	. . . . . . . . . . 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 13563	. . . . . . . . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 4111	. . . . . . . . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . . . . . . . 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 6120	. . . . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . . . 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 5665	. . . . . . . . . . . . . . . 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 5706	. . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . 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 4111	. . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . 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 6854	. . . . . . . . . . . 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 734	. . . . . . . . . . 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 6223	. . . . . . . . . . . . . . . . 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 5649	. . . . . . . . . . . . . . . . . . . 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 12888	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
370	70, 369	eqeltrrd 2853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝑦))
371		fzss2 13555	. . . . . . . . . . . . . . . . . . . . . . . 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 6015	. . . . . . . . . . . . . . . . . . . . . 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 13555	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
377	168	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
378	376, 377	ssneldd 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦))
379		eleq1 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦)))
380	379	notbid 320	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦)))
381	378, 380	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦)))
382	381	necon2ad 2962	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛 ≠ 𝑁))
383	382	imp 409	. . . . . . . . . . . . . . . . . . . . . . . 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 5183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
386	373, 385	eqtrd 2787	. . . . . . . . . . . . . . . . . . . . 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 5903	. . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))
390	389	elrnmpt 5923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
391	390	elv 3449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
392		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
393	392	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ)
394		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦))
395	118	jctl 530	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (1 ∈ ℤ ∧ 𝑦 ∈ ℤ))
396		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ)
397	396, 118	jctir 527	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
398		fzaddel 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
399	395, 397, 398	syl2an 604	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
400	394, 399	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
401		eleq1 2840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
402	400, 401	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
403	402	rexlimdva 3153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
404		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ)
405	404	zcnd 12664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ)
406		npcan1 11598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
409	408	ibir 270	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
410	409	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
411		peano2zm 12600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
412	404, 411	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈ ℤ)
413	412, 118	jctir 527	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
414		fzaddel 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
416	410, 415	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦))
417	405	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈ ℂ)
418	406	eqcomd 2758	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7388	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1))
421	420	rspceeqv 3595	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
422	416, 419, 421	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
423	422	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
424	403, 423	impbid 214	. . . . . . . . . . . . . . . . . . . . . 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	bitrid 285	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
427	426	eqrdv 2750	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1)))
428	388, 427	eqtrd 2787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1)))
429	428	imaeq2d 6035	. . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . 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 5665	. . . . . . . . . . . . . . 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 6120	. . . . . . . . . . . . . . . . . . 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 6223	. . . . . . . . . . . . . . . . . . . 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 6223	. . . . . . . . . . . . . . . . . . . 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 4110	. . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
438		fzsplit2 13540	. . . . . . . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
440	195	uneq2d 4112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
441	440	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
442	439, 441	eqtrd 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
443		uncom 4102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))
444	442, 443	eqtrdi 2803	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))))
445	444	imaeq2d 6035	. . . . . . . . . . . . . . . . . 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 4584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1})
447		fnsnfv 6931	. . . . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
450	449	imaeq2d 6035	. . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 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 5649	. . . . . . . . . . . . . . . . . . . . . 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 13554	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13555	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁))
459	458	resmptd 6015	. . . . . . . . . . . . . . . . . . . . . . . 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 13519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
461	166, 460	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))
462		eleq1 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
463	462	notbid 320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
464	461, 463	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁)
467	466	iffalsed 4481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
468	467	mpteq2dva 5183	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
469	468	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . . . . 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 5903	. . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . . 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 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ)
474	473	zcnd 12664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
478	477	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 12580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
480	117, 479	anim12ci 622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 527	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
483		fzaddel 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1))
488	420	rspceeqv 3595	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
489	485, 487, 488	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
490	489	ex 415	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
491		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))
492		elfzelz 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ)
493	492, 118	jctir 527	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
494		fzaddel 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
497		eleq1 2840	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
498	496, 497	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
499	498	rexlimdva 3153	. . . . . . . . . . . . . . . . . . . . . . . 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 214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
501		eqid 2752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))
502	501	elrnmpt 5923	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
503	502	elv 3449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
504	500, 503	bitr4di 291	. . . . . . . . . . . . . . . . . . . . . 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 2750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
506	69	oveq2d 7397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
507	506	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
508	472, 505, 507	3eqtr2rd 2794	. . . . . . . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . 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 5665	. . . . . . . . . . . . . . . 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 5706	. . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . 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 4115	. . . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . 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 6854	. . . . . . . . . . . 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 734	. . . . . . . . . . 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 2797	. . . . . . . . . 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 4734	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ)
524	138, 523	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {1} ⊆ ℂ
525		0cn 11157	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
526		snssi 4734	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
527	525, 526	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℂ
528	524, 527	unssi 4134	. . . . . . . . . . . . 13 ⊢ ({1} ∪ {0}) ⊆ ℂ
529	30	fconst 6735	. . . . . . . . . . . . . . . . 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 6735	. . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . 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 6711	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 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 6120	. . . . . . . . . . . . . . . . 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 13540	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
536	63, 370, 535	syl2an2 694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
537	536	imaeq2d 6035	. . . . . . . . . . . . . . . . . 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 6799	. . . . . . . . . . . . . . . . . . . 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 6768	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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	ffvelcdmda 7050	. . . . . . . . . . . . 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 3925	. . . . . . . . . . . 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	addlidd 11370	. . . . . . . . . . 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 483	. . . . . . . . . 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 2790	. . . . . . . . 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 4509	. . . . . . . 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 7397	. . . . . . 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 8815	. . . . . . . . . . . . 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	ffvelcdmda 7050	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
556		elfzonn0 13699	. . . . . . . . . . 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 12530	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
559	558	adantlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
560	138, 525	ifcli 4518	. . . . . . . . 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 11190	. . . . . . 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 2790	. . . . . 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 5183	. . . . 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 2787	. . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) = 0)
568		elfzle1 13518	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
569	568	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
570	567, 569	eqbrtrd 5112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) ≤ 𝑦)
571		0re 11169	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
572	566, 571	eqeltrdi 2860	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
573		lenlt 11247	. . . . . . . . 9 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
574	572, 231, 573	syl2an 604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
575	570, 574	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2^nd ‘𝑇))
576	575	iffalsed 4481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
577	576	csbeq1d 3847	. . . . 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 7414	. . . . . 6 ⊢ (𝑦 + 1) ∈ V
579		oveq2 7389	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
580	579	imaeq2d 6035	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
581	580	xpeq1d 5665	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
582		oveq1 7388	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
583	582	oveq1d 7396	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
584	583	imaeq2d 6035	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
585	584	xpeq1d 5665	. . . . . . . 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 4113	. . . . . . 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 7397	. . . . . 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 3878	. . . . 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 2803	. . . 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 7416	. . . . 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 6867	. . . . 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 2753	. . . . 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 6677	. . . . . 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 2914	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(2^nd ‘(1^st ‘𝑇))
595		nfmpt1 5189	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
596	594, 595	nfco 5826	. . . . . . . . . 10 ⊢ Ⅎ𝑛((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
597		nfcv 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1...𝑦)
598	596, 597	nfima 6043	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
599		nfcv 2914	. . . . . . . . 9 ⊢ Ⅎ𝑛{1}
600	598, 599	nfxp 5669	. . . . . . . 8 ⊢ Ⅎ𝑛((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1})
601		nfcv 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑦 + 1)...𝑁)
602	596, 601	nfima 6043	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
603		nfcv 2914	. . . . . . . . 9 ⊢ Ⅎ𝑛{0}
604	602, 603	nfxp 5669	. . . . . . . 8 ⊢ Ⅎ𝑛((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})
605	600, 604	nfun 4114	. . . . . . 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 6923	. . . . . 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 220	. . . . 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 7665	. . . 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 2798	. . 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 5183	. 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 2790	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 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 {cab 2730 ≠ wne 2947 ∀wral 3066 ∃wrex 3076 ∃!wreu 3355 {crab 3404 Vcvv 3444 ⦋csb 3843 ∖ cdif 3892 ∪ cun 3893 ∩ cin 3894 ⊆ wss 3895 ∅c0 4276 ifcif 4470 {csn 4572 ⟨cop 4578 class class class wbr 5090 ↦ cmpt 5171 × cxp 5634 ^◡ccnv 5635 ran crn 5637 ↾ cres 5638 “ cima 5639 ∘ ccom 5640 Fun wfun 6500 Fn wfn 6501 ⟶wf 6502 –onto→wfo 6504 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 ∘_f cof 7643 1^st c1st 7953 2^nd c2nd 7954 ↑_m cmap 8792 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 < clt 11202 ≤ cle 11203 − cmin 11400 ℕcn 12196 ℕ₀cn0 12467 ℤcz 12554 ℤ_≥cuz 12825 ...cfz 13498 ..^cfzo 13645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-er 8662 df-map 8794 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-nn 12197 df-n0 12468 df-z 12555 df-uz 12826 df-fz 13499 df-fzo 13646

This theorem is referenced by: poimirlem17 38074

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