Theorem poimirlem16 32595

Description: Lemma for poimir 32612 establishing the vertices of the simplex of poimirlem17 32596. (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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘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 → (2nd ‘𝑇) = 0)

Assertion

Ref	Expression
poimirlem16	⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘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 6103	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 4595	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4058	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3506	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
6		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇))
7	6	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
8	6	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
9	8	imaeq1d 5384	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
10	9	xpeq1d 5062	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
11	8	imaeq1d 5384	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
12	11	xpeq1d 5062	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
13	10, 12	uneq12d 3730	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
14	7, 13	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
15	14	csbeq2dv 3944	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
16	5, 15	eqtrd 2644	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
17	16	mpteq2dv 4673	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
18	17	eqeq2d 2620	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
19		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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
20	18, 19	elrab2 3333	. . . 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 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
21	20	simprbi 479	. . 3 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
22	1, 21	syl 17	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
23		elrabi 3328	. . . . . . . . . . . 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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
24	23, 19	eleq2s 2706	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
25	1, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
26		xp1st 7089	. . . . . . . . . 10 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
28		xp1st 7089	. . . . . . . . 9 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
30		elmapfn 7766	. . . . . . . 8 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
32	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
33		1ex 9914	. . . . . . . . . 10 ⊢ 1 ∈ V
34		fnconstg 6006	. . . . . . . . . 10 ⊢ (1 ∈ V → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))
35	33, 34	ax-mp 5	. . . . . . . . 9 ⊢ (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))
36		c0ex 9913	. . . . . . . . . 10 ⊢ 0 ∈ V
37		fnconstg 6006	. . . . . . . . . 10 ⊢ (0 ∈ V → (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
39	35, 38	pm3.2i 470	. . . . . . . 8 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
40		xp2nd 7090	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
41	27, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
42		fvex 6113	. . . . . . . . . . . . 13 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
43		f1oeq1 6040	. . . . . . . . . . . . 13 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
44	42, 43	elab 3319	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
45	41, 44	sylib 207	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
46		dff1o3 6056	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st ‘𝑇))))
47	46	simprbi 479	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st ‘𝑇)))
48	45, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun ◡(2nd ‘(1st ‘𝑇)))
49		imain 5888	. . . . . . . . . 10 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
51		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
52		nn0p1nn 11209	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
53	51, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
54	53	nnred 10912	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
55	54	ltp1d 10833	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
56		fzdisj 12239	. . . . . . . . . . . 12 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
57	55, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
58	57	imaeq2d 5385	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
59		ima0 5400	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
60	58, 59	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
61	50, 60	sylan9req 2665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
62		fnun 5911	. . . . . . . 8 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
63	39, 61, 62	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
64		imaundi 5464	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
65		nnuz 11599	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	53, 65	syl6eleq 2698	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ≥‘1))
67		peano2uz 11617	. . . . . . . . . . . . . 14 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
68	66, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
69	68	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
70		poimir.0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
71	70	nncnd 10913	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
72		npcan1 10334	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
73	71, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
74	73	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
75		elfzuz3 12210	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ≥‘𝑦))
76		eluzp1p1 11589	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
77	75, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
78	77	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
79	74, 78	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1)))
80		fzsplit2 12237	. . . . . . . . . . . 12 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
81	69, 79, 80	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
82	81	imaeq2d 5385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
83		f1ofo 6057	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
84		foima 6033	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
85	45, 83, 84	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
86	85	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
87	82, 86	eqtr3d 2646	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
88	64, 87	syl5eqr 2658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
89	88	fneq2d 5896	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
90	63, 89	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
91		ovex 6577	. . . . . . 7 ⊢ (1...𝑁) ∈ V
92	91	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
93		inidm 3784	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
94		eqidd 2611	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
95		eqidd 2611	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
96	32, 90, 92, 92, 93, 94, 95	offval 6802	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
97		oveq1 6556	. . . . . . . . . 10 ⊢ (1 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) → (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
98	97	eqeq2d 2620	. . . . . . . . 9 ⊢ (1 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) → ((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
99		oveq1 6556	. . . . . . . . . 10 ⊢ (0 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) → (0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
100	99	eqeq2d 2620	. . . . . . . . 9 ⊢ (0 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) → ((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
101		1p0e1 11010	. . . . . . . . . . . . . 14 ⊢ (1 + 0) = 1
102	101	eqcomi 2619	. . . . . . . . . . . . 13 ⊢ 1 = (1 + 0)
103		f1ofn 6051	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
104	45, 103	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
105	104	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
106		fzss2 12252	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
107	79, 106	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
108		eluzfz1 12219	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1)))
109	66, 108	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
110	109	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
111		fnfvima 6400	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))
112	105, 107, 110, 111	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))
113		fvun1 6179	. . . . . . . . . . . . . . . 16 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘1)))
114	35, 38, 113	mp3an12 1406	. . . . . . . . . . . . . . 15 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘1)))
115	61, 112, 114	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘1)))
116	33	fvconst2 6374	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘1)) = 1)
117	112, 116	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘1)) = 1)
118	115, 117	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = 1)
119		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
120	70	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
122	120, 121	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
123		1z 11284	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℤ
124	122, 123	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
125		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
126	125, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
127		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
128	124, 126, 127	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
129	119, 128	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
130	73	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
131	130	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
132	129, 131	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
133	132	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
134		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
135		peano2z 11295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
136	123, 135	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 + 1) ∈ ℤ
137	120, 136	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
138		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
139	138, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
140		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
141	137, 139, 140	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
142	134, 141	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
143		ax-1cn 9873	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℂ
144	143, 143	pncan3oi 10176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1 + 1) − 1) = 1
145	144	oveq1i 6559	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
146	142, 145	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
147	138	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
148		elfznn 12241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
149	148	nncnd 10913	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
150		subadd2 10164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
151	143, 150	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
152	151	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
153		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1))
154		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 − 1) = 𝑛 ↔ 𝑛 = (𝑦 − 1))
155	152, 153, 154	3bitr3g 301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
156	147, 149, 155	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
157	156	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
158	157	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
159		reu6i 3364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
160	146, 158, 159	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
161	160	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
162		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
163	162	f1ompt 6290	. . . . . . . . . . . . . . . . . . . . . . 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)))
164	133, 161, 163	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
165		f1osng 6089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
166	70, 33, 165	sylancl 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
167	70	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
168	167	ltm1d 10835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
169	122	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
170	169, 167	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
171	168, 170	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
172		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
173	171, 172	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
174		disjsn 4192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
175	173, 174	sylibr 223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
176		1re 9918	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
177	176	ltp1i 10806	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 < (1 + 1)
178	176, 176	readdcli 9932	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 1) ∈ ℝ
179	176, 178	ltnlei 10037	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
180	177, 179	mpbi 219	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ (1 + 1) ≤ 1
181		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
182	180, 181	mto 187	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
183		disjsn 4192	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
184	182, 183	mpbir 220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
185		f1oun 6069	. . . . . . . . . . . . . . . . . . . . . . 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}))
186	184, 185	mpanr2 716	. . . . . . . . . . . . . . . . . . . . . 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}))
187	164, 166, 175, 186	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
188		ssv 3588	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ℕ ⊆ V
189	188, 70	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ V)
190	33	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ V)
191	70, 65	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
192	73, 191	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
193		uzid 11578	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
194		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
195	122, 193, 194	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
196	73, 195	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
197		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
198	192, 196, 197	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
199	73	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
200		fzsn 12254	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
201	120, 200	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
202	199, 201	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
203	202	uneq2d 3729	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
204	198, 203	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
205		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
206	205	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
207	189, 190, 204, 206	fmptapd 6342	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
208		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
209	208	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
210	173, 209	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
211	210	necon2ad 2797	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁))
212	211	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁)
213		ifnefalse 4048	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
214	212, 213	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
215	214	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
216	215	uneq1d 3728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
217	207, 216	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
218	198, 203	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
219		uzid 11578	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ≥‘1))
220		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ (ℤ≥‘1) → (1 + 1) ∈ (ℤ≥‘1))
221	123, 219, 220	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 1) ∈ (ℤ≥‘1)
222		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
223	221, 191, 222	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
224		fzsn 12254	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℤ → (1...1) = {1})
225	123, 224	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...1) = {1}
226	225	uneq1i 3725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
227	226	equncomi 3721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
228	223, 227	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
229	217, 218, 228	f1oeq123d 6046	. . . . . . . . . . . . . . . . . . . . 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})))
230	187, 229	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
231		f1oco 6072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
232	45, 230, 231	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
233		dff1o3 6056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))))
234	233	simprbi 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
235	232, 234	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
236		imain 5888	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
237	235, 236	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
238	51	nn0red 11229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
239	238	ltp1d 10833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
240		fzdisj 12239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
241	239, 240	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
242	241	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅))
243		ima0 5400	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) = ∅
244	242, 243	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
245	237, 244	sylan9req 2665	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅)
246		imassrn 5396	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
247		f1of 6050	. . . . . . . . . . . . . . . . . . . . 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))
248		frn 5966	. . . . . . . . . . . . . . . . . . . . 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))
249	230, 247, 248	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁))
250	246, 249	syl5ss 3579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
251	250	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
252		elfz1end 12242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
253	70, 252	sylib 207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
254		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
255	205, 254, 33	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
256	253, 255	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
257	256	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
258		f1ofn 6051	. . . . . . . . . . . . . . . . . . . . . 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))
259	230, 258	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
260	259	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
261		fzss1 12251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
262	66, 261	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
263	262	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
264		eluzfz2 12220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
265	79, 264	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
266		fnfvima 6400	. . . . . . . . . . . . . . . . . . . 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)))
267	260, 263, 265, 266	syl3anc 1318	. . . . . . . . . . . . . . . . . . 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)))
268	257, 267	eqeltrrd 2689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
269		fnfvima 6400	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘(1st ‘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))) → ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
270	105, 251, 268, 269	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
271		imaco 5557	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
272	270, 271	syl6eleqr 2699	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘1) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
273		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
274	33, 273	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
275		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
276	36, 275	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
277		fvun2 6180	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) ∧ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘1) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘1)))
278	274, 276, 277	mp3an12 1406	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘1) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘1)))
279	245, 272, 278	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘1)))
280	36	fvconst2 6374	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇))‘1) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘1)) = 0)
281	272, 280	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘1)) = 0)
282	279, 281	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = 0)
283	282	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1))) = (1 + 0))
284	102, 118, 283	3eqtr4a 2670	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1))))
285		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘1) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)))
286		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘1) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)))
287	286	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘1) → (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1))))
288	285, 287	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘1) → ((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘1)))))
289	284, 288	syl5ibrcom 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st ‘𝑇))‘1) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
290	289	imp 444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
291	290	adantlr 747	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
292		eldifsn 4260	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘1)))
293		df-ne 2782	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘1) ↔ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1))
294	293	anbi2i 726	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘1)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)))
295	292, 294	bitri 263	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)))
296		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
297	33, 296	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))
298	297, 38	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
299		imain 5888	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
300	48, 299	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
301		fzdisj 12239	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
302	55, 301	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
303	302	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
304	303, 59	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
305	300, 304	sylan9req 2665	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
306		fnun 5911	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
307	298, 305, 306	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
308		imaundi 5464	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
309		fzpred 12259	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
310	191, 309	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
311		uncom 3719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
312	310, 311	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
313	312	difeq1d 3689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}))
314		difun2 4000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = (((1 + 1)...𝑁) ∖ {1})
315		disj3 3973	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}))
316	184, 315	mpbi 219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})
317	314, 316	eqtr4i 2635	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = ((1 + 1)...𝑁)
318	313, 317	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
319	318	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
320		eluzp1p1 11589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘(1 + 1)))
321	66, 320	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘(1 + 1)))
322	321	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘(1 + 1)))
323		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ≥‘(1 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
324	322, 79, 323	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
325	319, 324	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
326	325	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
327		imadif 5887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {1})))
328	48, 327	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {1})))
329		eluzfz1 12219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
330	191, 329	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ∈ (1...𝑁))
331		fnsnfv 6168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑇))‘1)} = ((2nd ‘(1st ‘𝑇)) “ {1}))
332	104, 330, 331	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘1)} = ((2nd ‘(1st ‘𝑇)) “ {1}))
333	332	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {1}) = {((2nd ‘(1st ‘𝑇))‘1)})
334	85, 333	difeq12d 3691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
335	328, 334	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
336	335	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
337	326, 336	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
338	308, 337	syl5eqr 2658	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
339	338	fneq2d 5896	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)})))
340	307, 339	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
341		incom 3767	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ({((2nd ‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))
342		disjdif 3992	. . . . . . . . . . . . . . . 16 ⊢ ({((2nd ‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)})) = ∅
343	341, 342	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅
344		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘1)})
345	33, 344	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘1)}
346		fvun1 6179	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
347	345, 346	mp3an2 1404	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
348		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘1)})
349	36, 348	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘1)}
350		fvun1 6179	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
351	349, 350	mp3an2 1404	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
352	347, 351	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd ‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
353	343, 352	mpanr1 715	. . . . . . . . . . . . . 14 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)})) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
354	340, 353	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘1)})) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
355	295, 354	sylan2br 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1))) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
356	355	anassrs 678	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
357		fzpred 12259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
358	66, 357	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
359	358	imaeq2d 5385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
360	359	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
361	332	uneq1d 3728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) = (((2nd ‘(1st ‘𝑇)) “ {1}) ∪ ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))))
362		uncom 3719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}) = ({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
363		imaundi 5464	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))) = (((2nd ‘(1st ‘𝑇)) “ {1}) ∪ ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
364	361, 362, 363	3eqtr4g 2669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}) = ((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
365	364	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}) = ((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
366	360, 365	eqtr4d 2647	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}))
367	366	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}) × {1}))
368		xpundir 5095	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st ‘𝑇))‘1)}) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))
369	367, 368	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1})))
370	369	uneq1d 3728	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1})) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
371		un23 3734	. . . . . . . . . . . . . 14 ⊢ (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1})) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))
372	370, 371	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1})))
373	372	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛))
374	373	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {1}))‘𝑛))
375		imaco 5557	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)))
376		df-ima 5051	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦))
377		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
378	75, 377	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
379	378	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
380	74, 379	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦))
381		fzss2 12252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...𝑁))
382	380, 381	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...𝑁))
383	382	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . 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))))
384	173	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
385		fzss2 12252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
386	75, 385	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
387	386	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
388	387	sseld 3567	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 ∈ (1...𝑦) → 𝑁 ∈ (1...(𝑁 − 1))))
389	384, 388	mtod 188	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦))
390		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦)))
391	390	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦)))
392	389, 391	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦)))
393	392	necon2ad 2797	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛 ≠ 𝑁))
394	393	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ≠ 𝑁)
395	394, 213	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
396	395	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
397	383, 396	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
398	397	rneqd 5274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
399	376, 398	syl5eq 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
400		vex 3176	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑗 ∈ V
401		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))
402	401	elrnmpt 5293	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
403	400, 402	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
404		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
405	404	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ)
406		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦))
407	123	jctl 562	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (1 ∈ ℤ ∧ 𝑦 ∈ ℤ))
408		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ)
409	408, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
410		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
411	407, 409, 410	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
412	406, 411	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
413		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
414	412, 413	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
415	414	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
416		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ)
417	416	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ)
418		npcan1 10334	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗)
419	417, 418	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
420	419	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
421	420	ibir 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
422	421	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
423		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
424	416, 423	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈ ℤ)
425	424, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
426		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
427	407, 425, 426	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
428	422, 427	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦))
429	417	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈ ℂ)
430	418	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℂ → 𝑗 = ((𝑗 − 1) + 1))
431	429, 430	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 = ((𝑗 − 1) + 1))
432		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1))
433	432	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = (𝑗 − 1) → (𝑗 = (𝑛 + 1) ↔ 𝑗 = ((𝑗 − 1) + 1)))
434	433	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
435	428, 431, 434	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
436	435	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
437	415, 436	impbid 201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
438	405, 437	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
439	403, 438	syl5bb 271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
440	439	eqrdv 2608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1)))
441	399, 440	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1)))
442	441	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦))) = ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
443	375, 442	syl5eq 2656	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))
444	443	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}))
445		imaundi 5464	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))))
446		imaco 5557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
447		imaco 5557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
448	446, 447	uneq12i 3727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
449	445, 448	eqtri 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
450	196	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
451		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
452	78, 450, 451	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
453	202	uneq2d 3729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
454	453	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
455	452, 454	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
456		uncom 3719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))
457	455, 456	syl6eq 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))))
458	457	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))))
459	256	sneqd 4137	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1})
460		fnsnfv 6168	. . . . . . . . . . . . . . . . . . . . . . . 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}))
461	259, 253, 460	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
462	459, 461	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
463	462	imaeq2d 5385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {1}) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
464	332, 463	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘1)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
465	464	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st ‘𝑇))‘1)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
466		df-ima 5051	. . . . . . . . . . . . . . . . . . . . . 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)))
467		fzss1 12251	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
468	66, 467	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
469		fzss2 12252	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
470	196, 469	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
471	468, 470	sylan9ssr 3582	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁))
472	471	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . 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))))
473		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
474	171, 473	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))
475		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
476	475	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
477	474, 476	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))))
478	477	con2d 128	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → ¬ 𝑛 = 𝑁))
479	478	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁)
480	479	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
481	480	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
482	481	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
483	472, 482	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 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)))
484	483	rneqd 5274	. . . . . . . . . . . . . . . . . . . . . 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)))
485	466, 484	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . 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)))
486		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ)
487	486	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℂ)
488	487, 418	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) = 𝑗)
489	488	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
490	489	ibir 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
491	490	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
492	53	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
493	122, 492	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
494	486, 423	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (𝑗 − 1) ∈ ℤ)
495	494, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
496		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
497	493, 495, 496	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
498	491, 497	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → (𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)))
499	487, 430	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 = ((𝑗 − 1) + 1))
500	499	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1))
501	433	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
502	498, 500, 501	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
503	502	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
504		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))
505		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ)
506	505, 123	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
507		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
508	493, 506, 507	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
509	504, 508	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
510		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
511	509, 510	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
512	511	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
513	503, 512	impbid 201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
514		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))
515	514	elrnmpt 5293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
516	400, 515	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
517	513, 516	syl6bbr 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))))
518	517	eqrdv 2608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
519	73	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
520	519	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
521	485, 518, 520	3eqtr2rd 2651	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
522	521	imaeq2d 5385	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
523	465, 522	uneq12d 3730	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))))
524	449, 458, 523	3eqtr4a 2670	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
525	524	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}))
526		xpundir 5095	. . . . . . . . . . . . . . . 16 ⊢ (({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}) = (({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
527	525, 526	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
528	444, 527	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
529		unass 3732	. . . . . . . . . . . . . . 15 ⊢ (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0})) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
530		un23 3734	. . . . . . . . . . . . . . 15 ⊢ (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0})) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))
531	529, 530	eqtr3i 2634	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))
532	528, 531	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0})))
533	532	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
534	533	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘1)} × {0}))‘𝑛))
535	356, 374, 534	3eqtr4d 2654	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
536		snssi 4280	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ)
537	143, 536	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {1} ⊆ ℂ
538		0cn 9911	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
539		snssi 4280	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
540	538, 539	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℂ
541	537, 540	unssi 3750	. . . . . . . . . . . . 13 ⊢ ({1} ∪ {0}) ⊆ ℂ
542	33	fconst 6004	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1}
543	36	fconst 6004	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}
544	542, 543	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0})
545		fun 5979	. . . . . . . . . . . . . . . 16 ⊢ (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}) ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
546	544, 245, 545	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
547		imaundi 5464	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
548	66	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ≥‘1))
549		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
550	548, 380, 549	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
551	550	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
552		f1ofo 6057	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁))
553		foima 6033	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
554	232, 552, 553	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
555	554	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
556	551, 555	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
557	547, 556	syl5eqr 2658	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
558	557	feq2d 5944	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
559	546, 558	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
560	559	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0}))
561	541, 560	sseldi 3566	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ)
562	561	addid2d 10116	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
563	562	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
564	535, 563	eqtr4d 2647	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘1)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
565	98, 100, 291, 564	ifbothda 4073	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
566	565	oveq2d 6565	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st ‘𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
567		elmapi 7765	. . . . . . . . . . . . 13 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
568	29, 567	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
569	568	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
570		elfzonn0 12380	. . . . . . . . . . 11 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
571	569, 570	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
572	571	nn0cnd 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
573	572	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
574	143, 538	keepel 4105	. . . . . . . . 9 ⊢ if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) ∈ ℂ
575	574	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) ∈ ℂ)
576	573, 575, 561	addassd 9941	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st ‘𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
577	566, 576	eqtr4d 2647	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
578	577	mpteq2dva 4672	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
579	96, 578	eqtrd 2644	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
580		poimirlem18.4	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑇) = 0)
581	580	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑇) = 0)
582		elfzle1 12215	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
583	582	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
584	581, 583	eqbrtrd 4605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑇) ≤ 𝑦)
585		0re 9919	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
586	580, 585	syl6eqel 2696	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℝ)
587		lenlt 9995	. . . . . . . . 9 ⊢ (((2nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇)))
588	586, 238, 587	syl2an 493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇)))
589	584, 588	mpbid 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd ‘𝑇))
590	589	iffalsed 4047	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
591	590	csbeq1d 3506	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
592		ovex 6577	. . . . . 6 ⊢ (𝑦 + 1) ∈ V
593		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
594	593	imaeq2d 5385	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))
595	594	xpeq1d 5062	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
596		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
597	596	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
598	597	imaeq2d 5385	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
599	598	xpeq1d 5062	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
600	595, 599	uneq12d 3730	. . . . . . 7 ⊢ (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
601	600	oveq2d 6565	. . . . . 6 ⊢ (𝑗 = (𝑦 + 1) → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
602	592, 601	csbie 3525	. . . . 5 ⊢ ⦋(𝑦 + 1) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
603	591, 602	syl6eq 2660	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
604		ovex 6577	. . . . . 6 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) ∈ V
605	604	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) ∈ V)
606		fvex 6113	. . . . . 6 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V
607	606	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V)
608		eqidd 2611	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))))
609		ffn 5958	. . . . . . 7 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
610	559, 609	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
611		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(2nd ‘(1st ‘𝑇))
612		nfmpt1 4675	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
613	611, 612	nfco 5209	. . . . . . . . . 10 ⊢ Ⅎ𝑛((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
614		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1...𝑦)
615	613, 614	nfima 5393	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
616		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛{1}
617	615, 616	nfxp 5066	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1})
618		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑦 + 1)...𝑁)
619	613, 618	nfima 5393	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
620		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛{0}
621	619, 620	nfxp 5066	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})
622	617, 621	nfun 3731	. . . . . . 7 ⊢ Ⅎ𝑛(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))
623	622	dffn5f 6162	. . . . . 6 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
624	610, 623	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
625	92, 605, 607, 608, 624	offval2 6812	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
626	579, 603, 625	3eqtr4rd 2655	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
627	626	mpteq2dva 4672	. 2 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
628	22, 627	eqtr4d 2647	1 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335

This theorem is referenced by:  poimirlem17  32596