Theorem poimirlem19 32598

Description: Lemma for poimir 32612 establishing the vertices of the simplex in poimirlem20 32599. (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)
poimirlem22.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
poimirlem21.4	⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)

Assertion

Ref	Expression
poimirlem19	⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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 poimirlem19

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}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)))
35	33, 34	ax-mp 5	. . . . . . . . 9 ⊢ (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦))
36		c0ex 9913	. . . . . . . . . 10 ⊢ 0 ∈ V
37		fnconstg 6006	. . . . . . . . . 10 ⊢ (0 ∈ V → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))
39	35, 38	pm3.2i 470	. . . . . . . 8 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 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) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
51		elfznn0 12302	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
52	51	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
53	52	ltp1d 10833	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
54		fzdisj 12239	. . . . . . . . . . . 12 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
55	53, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
56	55	imaeq2d 5385	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
57		ima0 5400	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
58	56, 57	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
59	50, 58	sylan9req 2665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
60		fnun 5911	. . . . . . . 8 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
61	39, 59, 60	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
62		imaundi 5464	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
63		nn0p1nn 11209	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
64	51, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
65		nnuz 11599	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
66	64, 65	syl6eleq 2698	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ≥‘1))
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ≥‘1))
68		poimir.0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nncnd 10913	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
70		npcan1 10334	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
71	69, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
72	71	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
73		elfzuz3 12210	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ≥‘𝑦))
74		peano2uz 11617	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
75	73, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
76	75	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
77	72, 76	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦))
78		fzsplit2 12237	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
79	67, 77, 78	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
80	79	imaeq2d 5385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
81		f1ofo 6057	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
82		foima 6033	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
83	45, 81, 82	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
84	83	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
85	80, 84	eqtr3d 2646	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
86	62, 85	syl5eqr 2658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
87	86	fneq2d 5896	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
88	61, 87	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
89		ovex 6577	. . . . . . 7 ⊢ (1...𝑁) ∈ V
90	89	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
91		inidm 3784	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
92		eqidd 2611	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
93		eqidd 2611	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
94	32, 88, 90, 90, 91, 92, 93	offval 6802	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
95		elmapi 7765	. . . . . . . . . . . . 13 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
96	29, 95	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
97	96	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
98		elfzonn0 12380	. . . . . . . . . . 11 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
99	97, 98	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
100	99	nn0cnd 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
101	100	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
102		ax-1cn 9873	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
103		0cn 9911	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
104	102, 103	keepel 4105	. . . . . . . . 9 ⊢ if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) ∈ ℂ
105	104	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) ∈ ℂ)
106		snssi 4280	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ)
107	102, 106	ax-mp 5	. . . . . . . . . 10 ⊢ {1} ⊆ ℂ
108		snssi 4280	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
109	103, 108	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ ℂ
110	107, 109	unssi 3750	. . . . . . . . 9 ⊢ ({1} ∪ {0}) ⊆ ℂ
111	33	fconst 6004	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1}
112	36	fconst 6004	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}
113	111, 112	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0})
114		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
115	68	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
116		1z 11284	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
117		peano2z 11295	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
118	116, 117	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) ∈ ℤ
119	115, 118	jctil 558	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
120		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
121	120, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
122		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
123	119, 121, 122	syl2an 493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
124	114, 123	mpbid 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
125	102, 102	pncan3oi 10176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 + 1) − 1) = 1
126	125	oveq1i 6559	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
127	124, 126	syl6eleq 2698	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
128	127	ralrimiva 2949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
129		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
130		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
131	115, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
132	131, 116	jctil 558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
133		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
134	133, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
135		fzaddel 12246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
136	132, 134, 135	syl2an 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
137	129, 136	mpbid 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
138	71	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
139	138	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
140	137, 139	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
141	120	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
142	133	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
143		subadd2 10164	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
144	102, 143	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
145		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
146		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
147	144, 145, 146	3bitr4g 302	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
148	141, 142, 147	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
149	148	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
150	149	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
151		reu6i 3364	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
152	140, 150, 151	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
153	152	ralrimiva 2949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
154		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
155	154	f1ompt 6290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)))
156	128, 153, 155	sylanbrc 695	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
157		f1osng 6089	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
158	33, 68, 157	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
159	68	nnred 10912	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ ℝ)
160	159	ltm1d 10835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
161	131	zred 11358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
162	161, 159	ltnled 10063	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
163	160, 162	mpbid 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
164		elfzle2 12216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
165	163, 164	nsyl 134	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
166		disjsn 4192	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
167	165, 166	sylibr 223	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
168		1re 9918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℝ
169	168	ltp1i 10806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 < (1 + 1)
170	118	zrei 11260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) ∈ ℝ
171	168, 170	ltnlei 10037	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
172	169, 171	mpbi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ (1 + 1) ≤ 1
173		elfzle1 12215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
174	172, 173	mto 187	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
175		disjsn 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
176	174, 175	mpbir 220	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
177		f1oun 6069	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
178	176, 177	mpanr1 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
179	156, 158, 167, 178	syl21anc 1317	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
180		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
181	174, 180	mtbiri 316	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
182	181	necon2ai 2811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
183		ifnefalse 4048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
184	182, 183	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
185	184	mpteq2ia 4668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
186	185	uneq1i 3725	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
187	33	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ V)
188		ssv 3588	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ ⊆ V
189	188, 68	sseldi 3566	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ V)
190	68, 65	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
191		fzpred 12259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
192	190, 191	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
193		uncom 3719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
194	192, 193	syl6req 2661	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
195		iftrue 4042	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
196	195	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
197	187, 189, 194, 196	fmptapd 6342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
198	186, 197	syl5eqr 2658	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
199	71, 190	eqeltrd 2688	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
200		uzid 11578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
201		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
202	131, 200, 201	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
203	71, 202	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
204		fzsplit2 12237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
205	199, 203, 204	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
206	71	oveq1d 6564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
207		fzsn 12254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
208	115, 207	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
209	206, 208	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
210	209	uneq2d 3729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
211	205, 210	eqtr2d 2645	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
212	198, 194, 211	f1oeq123d 6046	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)))
213	179, 212	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
214		f1oco 6072	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
215	45, 213, 214	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
216		dff1o3 6056	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))))
217	216	simprbi 479	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
218		imain 5888	. . . . . . . . . . . . . 14 ⊢ (Fun ◡((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
219	215, 217, 218	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
220	64	nnred 10912	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
221	220	ltp1d 10833	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
222		fzdisj 12239	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
223	221, 222	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
224	223	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅))
225		ima0 5400	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅) = ∅
226	224, 225	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
227	219, 226	sylan9req 2665	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
228		fun 5979	. . . . . . . . . . . 12 ⊢ (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}) ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
229	113, 227, 228	sylancr 694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
230		imaundi 5464	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
231	64	peano2nnd 10914	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
232	231, 65	syl6eleq 2698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
233	232	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
234		eluzp1p1 11589	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
235	73, 234	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
236	235	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
237	72, 236	eqeltrrd 2689	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1)))
238		fzsplit2 12237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
239	233, 237, 238	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
240	239	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
241		f1ofo 6057	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁))
242		foima 6033	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
243	215, 241, 242	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
244	243	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
245	240, 244	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
246	230, 245	syl5eqr 2658	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
247	246	feq2d 5944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
248	229, 247	mpbid 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
249	248	ffvelrnda 6267	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0}))
250	110, 249	sseldi 3566	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ)
251	101, 105, 250	subadd23d 10293	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st ‘𝑇))‘𝑛) + (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))))
252		oveq2 6557	. . . . . . . . . 10 ⊢ (1 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)))
253	252	eqeq1d 2612	. . . . . . . . 9 ⊢ (1 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
254		oveq2 6557	. . . . . . . . . 10 ⊢ (0 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)))
255	254	eqeq1d 2612	. . . . . . . . 9 ⊢ (0 = if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
256		1m1e0 10966	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
257		f1ofn 6051	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
258	45, 257	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
259	258	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
260		imassrn 5396	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
261		f1of 6050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
262	213, 261	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
263		frn 5966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁) → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
264	262, 263	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
265	260, 264	syl5ss 3579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
266	265	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
267		eqidd 2611	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
268		eluzfz1 12219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
269	190, 268	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ (1...𝑁))
270	267, 196, 269, 68	fvmptd 6197	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
271	270	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
272		f1ofn 6051	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
273	213, 272	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
274	273	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
275		fzss2 12252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
276	237, 275	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
277		eluzfz1 12219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1)))
278	66, 277	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
279	278	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
280		fnfvima 6400	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
281	274, 276, 279, 280	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
282	271, 281	eqeltrrd 2689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
283		fnfvima 6400	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
284	259, 266, 282, 283	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
285		imaco 5557	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
286	284, 285	syl6eleqr 2699	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
287		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
288	33, 287	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
289		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
290	36, 289	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
291		fvun1 6179	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∧ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) ∧ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
292	288, 290, 291	mp3an12 1406	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
293	227, 286, 292	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
294	33	fvconst2 6374	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇))‘𝑁) ∈ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 1)
295	286, 294	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 1)
296	293, 295	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 1)
297	296	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) − 1) = (1 − 1))
298		fzss1 12251	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
299	66, 298	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
300	299	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
301		eluzfz2 12220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
302	237, 301	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
303		fnfvima 6400	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
304	259, 300, 302, 303	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
305		fvun2 6180	. . . . . . . . . . . . . . . 16 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
306	35, 38, 305	mp3an12 1406	. . . . . . . . . . . . . . 15 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
307	59, 304, 306	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)))
308	36	fvconst2 6374	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
309	304, 308	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
310	307, 309	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
311	256, 297, 310	3eqtr4a 2670	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)))
312		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)))
313	312	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) − 1))
314		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)))
315	313, 314	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁) → ((((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁))))
316	311, 315	syl5ibrcom 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
317	316	imp 444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
318	317	adantlr 747	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
319	250	subid1d 10260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
320	319	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
321		eldifsn 4260	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁)))
322		df-ne 2782	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁) ↔ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁))
323	322	anbi2i 726	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)))
324	321, 323	bitri 263	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)))
325		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
326	36, 325	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))
327	35, 326	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
328		imain 5888	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
329	48, 328	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
330		fzdisj 12239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
331	53, 330	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
332	331	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
333	332, 57	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
334	329, 333	sylan9req 2665	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
335		fnun 5911	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ∧ (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
336	327, 334, 335	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
337		imaundi 5464	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
338	205, 210	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
339	338	difeq1d 3689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
340		difun2 4000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
341	339, 340	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}))
342		difsn 4269	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
343	165, 342	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
344	341, 343	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
345	344	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
346	73	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑦))
347		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 1) ∈ (ℤ≥‘1) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑦)) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
348	67, 346, 347	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
349	345, 348	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
350	349	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))))
351		imadif 5887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})))
352	48, 351	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})))
353		elfz1end 12242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
354	68, 353	sylib 207	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
355		fnsnfv 6168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
356	258, 354, 355	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
357	356	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {𝑁}) = {((2nd ‘(1st ‘𝑇))‘𝑁)})
358	83, 357	difeq12d 3691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
359	352, 358	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
360	359	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
361	350, 360	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
362	337, 361	syl5eqr 2658	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
363	362	fneq2d 5896	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ↔ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)})))
364	336, 363	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
365		incom 3767	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ({((2nd ‘(1st ‘𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
366		disjdif 3992	. . . . . . . . . . . . . . . 16 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)})) = ∅
367	365, 366	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅
368		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)})
369	33, 368	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)}
370		fvun1 6179	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
371	369, 370	mp3an2 1404	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
372		fnconstg 6006	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)})
373	36, 372	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)}
374		fvun1 6179	. . . . . . . . . . . . . . . . 17 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
375	373, 374	mp3an2 1404	. . . . . . . . . . . . . . . 16 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
376	371, 375	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
377	367, 376	mpanr1 715	. . . . . . . . . . . . . 14 ⊢ ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)})) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
378	364, 377	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)})) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
379	324, 378	sylan2br 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁))) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
380	379	anassrs 678	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
381		imaundi 5464	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}))
382		imaco 5557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
383		imaco 5557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
384	382, 383	uneq12i 3727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1})) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
385	381, 384	eqtri 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
386		fzpred 12259	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
387	66, 386	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
388		uncom 3719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({1} ∪ ((1 + 1)...(𝑦 + 1))) = (((1 + 1)...(𝑦 + 1)) ∪ {1})
389	387, 388	syl6eq 2660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = (((1 + 1)...(𝑦 + 1)) ∪ {1}))
390	389	imaeq2d 5385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
391	390	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
392		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
393	125	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → ((1 + 1) − 1) = 1)
394		zcn 11259	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
395		pncan1 10333	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
396	394, 395	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → ((𝑦 + 1) − 1) = 𝑦)
397	393, 396	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = (1...𝑦))
398		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℤ)
399	398	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℂ)
400		pncan1 10333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
401	399, 400	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) = 𝑗)
402	401	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
403	402	ibir 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
404	403	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
405		peano2z 11295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
406	405, 118	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ℤ → ((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ))
407	398	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (𝑗 + 1) ∈ ℤ)
408	407, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
409		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
410	406, 408, 409	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
411	404, 410	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → (𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
412	401	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 = ((𝑗 + 1) − 1))
413	412	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → 𝑗 = ((𝑗 + 1) − 1))
414		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = (𝑗 + 1) → (𝑛 − 1) = ((𝑗 + 1) − 1))
415	414	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = (𝑗 + 1) → (𝑗 = (𝑛 − 1) ↔ 𝑗 = ((𝑗 + 1) − 1)))
416	415	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
417	411, 413, 416	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
418	417	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
419		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
420		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ∈ ℤ)
421	420, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
422		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
423	406, 421, 422	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
424	419, 423	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
425		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
426	424, 425	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
427	426	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
428	418, 427	impbid 201	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
429		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑗 ∈ V
430		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
431	430	elrnmpt 5293	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
432	429, 431	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
433	428, 432	syl6bbr 277	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))))
434	433	eqrdv 2608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
435	397, 434	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
436	392, 435	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
437	436	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
438		df-ima 5051	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1)))
439		uzid 11578	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ≥‘1))
440		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ (ℤ≥‘1) → (1 + 1) ∈ (ℤ≥‘1))
441	116, 439, 440	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 1) ∈ (ℤ≥‘1)
442		fzss1 12251	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1 + 1) ∈ (ℤ≥‘1) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1)))
443	441, 442	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1))
444	443, 276	syl5ss 3579	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...𝑁))
445	444	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
446		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ ((1 + 1)...(𝑦 + 1)) → (1 + 1) ≤ 1)
447	172, 446	mto 187	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ¬ 1 ∈ ((1 + 1)...(𝑦 + 1))
448		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ 1 ∈ ((1 + 1)...(𝑦 + 1))))
449	447, 448	mtbiri 316	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
450	449	necon2ai 2811	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ≠ 1)
451	450, 183	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
452	451	mpteq2ia 4668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
453	445, 452	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
454	453	rneqd 5274	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
455	438, 454	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
456	437, 455	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
457	456	imaeq2d 5385	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))))
458	270	sneqd 4137	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = {𝑁})
459		fnsnfv 6168	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
460	273, 269, 459	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
461	458, 460	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑁} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
462	461	imaeq2d 5385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {𝑁}) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
463	356, 462	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
464	463	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
465	457, 464	uneq12d 3730	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))))
466	385, 391, 465	3eqtr4a 2670	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
467	466	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) × {1}))
468		xpundir 5095	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))
469	467, 468	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1})))
470		imaco 5557	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)))
471		df-ima 5051	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁))
472		fzss1 12251	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ≥‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
473	233, 472	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
474	473	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
475		1red 9934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ ℝ)
476	64	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
477	476	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℤ)
478	477	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
479	64	nnge1d 10940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ≤ (𝑦 + 1))
480	475, 220, 478, 479, 221	lelttrd 10074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 < ((𝑦 + 1) + 1))
481	475, 478	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1 < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ 1))
482	480, 481	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ ((𝑦 + 1) + 1) ≤ 1)
483		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ 1)
484	482, 483	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁))
485		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 1 → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
486	485	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 1 → (¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
487	484, 486	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 = 1 → ¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)))
488	487	necon2ad 2797	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ≠ 1))
489	488	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ≠ 1)
490	489, 183	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
491	490	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
492	491	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
493	474, 492	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
494	493	rneqd 5274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
495	471, 494	syl5eq 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
496		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))
497	496	elrnmpt 5293	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
498	429, 497	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
499		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁))
500	115, 477	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
501		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ∈ ℤ)
502	501, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
503		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
504	500, 502, 503	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
505	499, 504	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
506		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
507	505, 506	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
508	507	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
509		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℤ)
510	509	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℂ)
511	510, 400	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) = 𝑗)
512	511	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
513	512	ibir 256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
514	513	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
515	509	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (𝑗 + 1) ∈ ℤ)
516	515, 116	jctir 559	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
517		fzsubel 12248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
518	500, 516, 517	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
519	514, 518	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → (𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
520	511	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 = ((𝑗 + 1) − 1))
521	520	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → 𝑗 = ((𝑗 + 1) − 1))
522	415	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
523	519, 521, 522	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
524	523	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
525	508, 524	impbid 201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
526	498, 525	syl5bb 271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
527	526	eqrdv 2608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
528	64	nncnd 10913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℂ)
529		pncan1 10333	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ ℂ → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
530	528, 529	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
531	530	oveq1d 6564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
532	531	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
533	495, 527, 532	3eqtrd 2648	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ((𝑦 + 1)...(𝑁 − 1)))
534	533	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
535	470, 534	syl5eq 2656	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
536	535	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))
537	469, 536	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})))
538		un23 3734	. . . . . . . . . . . . . 14 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))
539	537, 538	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1})))
540	539	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛))
541	540	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛))
542		imaundi 5464	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
543		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
544	235, 203, 543	syl2anr 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
545	209	uneq2d 3729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
546	545	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
547	544, 546	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
548	547	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})))
549	356	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
550	549	uneq2d 3729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ {𝑁})))
551	542, 548, 550	3eqtr4a 2670	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
552	551	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) × {0}))
553		xpundir 5095	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) × {0}) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))
554	552, 553	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})))
555	554	uneq2d 3729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))))
556		unass 3732	. . . . . . . . . . . . . 14 ⊢ (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})))
557	555, 556	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})))
558	557	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
559	558	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛))
560	380, 541, 559	3eqtr4d 2654	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
561	320, 560	eqtrd 2644	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
562	253, 255, 318, 561	ifbothda 4073	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
563	562	oveq2d 6565	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + (((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) = (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
564	251, 563	eqtr2d 2645	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
565	564	mpteq2dva 4672	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
566	94, 565	eqtrd 2644	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
567	52	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
568	161	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
569	159	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
570		elfzle2 12216	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
571	570	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
572	160	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
573	567, 568, 569, 571, 572	lelttrd 10074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
574		poimirlem21.4	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)
575	574	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑇) = 𝑁)
576	573, 575	breqtrrd 4611	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd ‘𝑇))
577	576	iftrued 4044	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
578	577	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 / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘𝑓 + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
579		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
580		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
581	580	imaeq2d 5385	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)))
582	581	xpeq1d 5062	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}))
583		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
584	583	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
585	584	imaeq2d 5385	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
586	585	xpeq1d 5062	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
587	582, 586	uneq12d 3730	. . . . . . 7 ⊢ (𝑗 = 𝑦 → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
588	587	oveq2d 6565	. . . . . 6 ⊢ (𝑗 = 𝑦 → ((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}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
589	579, 588	csbie 3525	. . . . 5 ⊢ ⦋𝑦 / 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}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
590	578, 589	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}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
591		ovex 6577	. . . . . 6 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) ∈ V
592	591	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) ∈ V)
593		fvex 6113	. . . . . 6 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V
594	593	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V)
595		eqidd 2611	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))))
596		ffn 5958	. . . . . . 7 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
597	248, 596	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
598		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(2nd ‘(1st ‘𝑇))
599		nfmpt1 4675	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
600	598, 599	nfco 5209	. . . . . . . . . 10 ⊢ Ⅎ𝑛((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
601		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1...(𝑦 + 1))
602	600, 601	nfima 5393	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
603		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛{1}
604	602, 603	nfxp 5066	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})
605		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛(((𝑦 + 1) + 1)...𝑁)
606	600, 605	nfima 5393	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
607		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛{0}
608	606, 607	nfxp 5066	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})
609	604, 608	nfun 3731	. . . . . . 7 ⊢ Ⅎ𝑛(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
610	609	dffn5f 6162	. . . . . 6 ⊢ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
611	597, 610	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
612	90, 592, 594, 595, 611	offval2 6812	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
613	566, 590, 612	3eqtr4rd 2655	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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}))))
614	613	mpteq2dva 4672	. 2 ⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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})))))
615	22, 614	eqtr4d 2647	1 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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:  poimirlem20  32599