Theorem poimirlem19 33428

Description: Lemma for poimir 33442 establishing the vertices of the simplex in poimirlem20 33429. (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 6191	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
3	2	breq2d 4665	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
4	3	ifbid 4108	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
5	4	csbeq1d 3540	. . . . . . . 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 6191	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇))
7	6	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
8	6	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
9	8	imaeq1d 5465	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
10	9	xpeq1d 5138	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
11	8	imaeq1d 5465	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
12	11	xpeq1d 5138	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
13	10, 12	uneq12d 3768	. . . . . . . . . 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 6668	. . . . . . . . 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 3992	. . . . . . . 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 2656	. . . . . . 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 4745	. . . . . 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 2632	. . . . 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 3366	. . . 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 480	. . 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 3359	. . . . . . . . . . . 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 2719	. . . . . . . . . . 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 7198	. . . . . . . . . 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 7198	. . . . . . . . 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 7880	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st ‘𝑇)) Fn (1...𝑁))
33		1ex 10035	. . . . . . . . . 10 ⊢ 1 ∈ V
34		fnconstg 6093	. . . . . . . . . 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 10034	. . . . . . . . . 10 ⊢ 0 ∈ V
37		fnconstg 6093	. . . . . . . . . 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 471	. . . . . . . 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 7199	. . . . . . . . . . . . 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 6201	. . . . . . . . . . . . 13 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
43		f1oeq1 6127	. . . . . . . . . . . . 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 3350	. . . . . . . . . . . 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 208	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
46		dff1o3 6143	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 5974	. . . . . . . . . 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 12433	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
52	51	nn0red 11352	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
53	52	ltp1d 10954	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
54		fzdisj 12368	. . . . . . . . . . . 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 5466	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
57		ima0 5481	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)) “ ∅) = ∅
58	56, 57	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
59	50, 58	sylan9req 2677	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
60		fnun 5997	. . . . . . . 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 695	. . . . . . 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 5545	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
63		nn0p1nn 11332	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
64	51, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
65		nnuz 11723	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
66	64, 65	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ≥‘1))
67	66	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ≥‘1))
68		poimir.0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nncnd 11036	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
70		npcan1 10455	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
71	69, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
72	71	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
73		elfzuz3 12339	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ≥‘𝑦))
74		peano2uz 11741	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑦))
77	72, 76	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦))
78		fzsplit2 12366	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
79	67, 77, 78	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
80	79	imaeq2d 5466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
81		f1ofo 6144	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
82		foima 6120	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
85	80, 84	eqtr3d 2658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
86	62, 85	syl5eqr 2670	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
87	86	fneq2d 5982	. . . . . . 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 222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
89		ovexd 6680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
90		inidm 3822	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
91		eqidd 2623	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st ‘𝑇))‘𝑛))
92		eqidd 2623	. . . . . 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))
93	32, 88, 89, 89, 90, 91, 92	offval 6904	. . . . 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))))
94		elmapi 7879	. . . . . . . . . . . . 13 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
95	29, 94	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
96	95	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
97		elfzonn0 12512	. . . . . . . . . . 11 ⊢ (((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℕ0)
99	98	nn0cnd 11353	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
100	99	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st ‘𝑇))‘𝑛) ∈ ℂ)
101		ax-1cn 9994	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
102		0cn 10032	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
103	101, 102	keepel 4155	. . . . . . . . 9 ⊢ if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) ∈ ℂ
104	103	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0) ∈ ℂ)
105		snssi 4339	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ)
106	101, 105	ax-mp 5	. . . . . . . . . 10 ⊢ {1} ⊆ ℂ
107		snssi 4339	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
108	102, 107	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ ℂ
109	106, 108	unssi 3788	. . . . . . . . 9 ⊢ ({1} ∪ {0}) ⊆ ℂ
110	33	fconst 6091	. . . . . . . . . . . . 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}
111	36	fconst 6091	. . . . . . . . . . . . 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}
112	110, 111	pm3.2i 471	. . . . . . . . . . . 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})
113		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
114	68	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
115		1z 11407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
116		peano2z 11418	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
117	115, 116	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) ∈ ℤ
118	114, 117	jctil 560	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
119		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
120	119, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
121		fzsubel 12377	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
122	118, 120, 121	syl2an 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
123	113, 122	mpbid 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
124	101, 101	pncan3oi 10297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 + 1) − 1) = 1
125	124	oveq1i 6660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
126	123, 125	syl6eleq 2711	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
127	126	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
128		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
129		peano2zm 11420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
130	114, 129	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
131	130, 115	jctil 560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
132		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
133	132, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
134		fzaddel 12375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
135	131, 133, 134	syl2an 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
136	128, 135	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
137	71	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
138	137	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
139	136, 138	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
140	119	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
141	132	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
142		subadd2 10285	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
143	101, 142	mp3an2 1412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
144		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
145		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
146	143, 144, 145	3bitr4g 303	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
147	140, 141, 146	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
148	147	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
149	148	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
150		reu6i 3397	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
151	139, 149, 150	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
152	151	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
153		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
154	153	f1ompt 6382	. . . . . . . . . . . . . . . . . 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)))
155	127, 152, 154	sylanbrc 698	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
156		f1osng 6177	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
157	33, 68, 156	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
158	68	nnred 11035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ ℝ)
159	158	ltm1d 10956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
160	130	zred 11482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
161	160, 158	ltnled 10184	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
162	159, 161	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
163		elfzle2 12345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
164	162, 163	nsyl 135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
165		disjsn 4246	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
166	164, 165	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
167		1re 10039	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℝ
168	167	ltp1i 10927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 < (1 + 1)
169	117	zrei 11383	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) ∈ ℝ
170	167, 169	ltnlei 10158	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
171	168, 170	mpbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ (1 + 1) ≤ 1
172		elfzle1 12344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
173	171, 172	mto 188	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
174		disjsn 4246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
175	173, 174	mpbir 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
176		f1oun 6156	. . . . . . . . . . . . . . . . . 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}))
177	175, 176	mpanr1 719	. . . . . . . . . . . . . . . . 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}))
178	155, 157, 166, 177	syl21anc 1325	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
179		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
180	173, 179	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
181	180	necon2ai 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
182		ifnefalse 4098	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
183	181, 182	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
184	183	mpteq2ia 4740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
185	184	uneq1i 3763	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
186	33	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ V)
187		ssv 3625	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ ⊆ V
188	187, 68	sseldi 3601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ V)
189	68, 65	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
190		fzpred 12389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
191	189, 190	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
192		uncom 3757	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
193	191, 192	syl6req 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
194		iftrue 4092	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
195	194	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
196	186, 188, 193, 195	fmptapd 6437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
197	185, 196	syl5eqr 2670	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
198	71, 189	eqeltrd 2701	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
199		uzid 11702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
200		peano2uz 11741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
201	130, 199, 200	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
202	71, 201	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
203		fzsplit2 12366	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
204	198, 202, 203	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
205	71	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
206		fzsn 12383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
207	114, 206	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
208	205, 207	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
209	208	uneq2d 3767	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
210	204, 209	eqtr2d 2657	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
211	197, 193, 210	f1oeq123d 6133	. . . . . . . . . . . . . . . 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)))
212	178, 211	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
213		f1oco 6159	. . . . . . . . . . . . . . 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))
214	45, 212, 213	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
215		dff1o3 6143	. . . . . . . . . . . . . . 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))))))
216	215	simprbi 480	. . . . . . . . . . . . . 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)))))
217		imain 5974	. . . . . . . . . . . . . 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))))
218	214, 216, 217	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))))
219	64	nnred 11035	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
220	219	ltp1d 10954	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
221		fzdisj 12368	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
222	220, 221	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
223	222	imaeq2d 5466	. . . . . . . . . . . . . 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)))) “ ∅))
224		ima0 5481	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅) = ∅
225	223, 224	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
226	218, 225	sylan9req 2677	. . . . . . . . . . . 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))) = ∅)
227		fun 6066	. . . . . . . . . . . 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}))
228	112, 226, 227	sylancr 695	. . . . . . . . . . 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}))
229		imaundi 5545	. . . . . . . . . . . . 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)))
230	64	peano2nnd 11037	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
231	230, 65	syl6eleq 2711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
232	231	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ≥‘1))
233		eluzp1p1 11713	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
234	73, 233	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
235	234	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)))
236	72, 235	eqeltrrd 2702	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1)))
237		fzsplit2 12366	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
238	232, 236, 237	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
239	238	imaeq2d 5466	. . . . . . . . . . . . . 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))))
240		f1ofo 6144	. . . . . . . . . . . . . . . 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))
241		foima 6120	. . . . . . . . . . . . . . . 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))
242	214, 240, 241	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
243	242	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
244	239, 243	eqtr3d 2658	. . . . . . . . . . . . 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))
245	229, 244	syl5eqr 2670	. . . . . . . . . . . 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))
246	245	feq2d 6031	. . . . . . . . . . 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})))
247	228, 246	mpbid 222	. . . . . . . . . 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}))
248	247	ffvelrnda 6359	. . . . . . . . 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}))
249	109, 248	sseldi 3601	. . . . . . . 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)
250	100, 104, 249	subadd23d 10414	. . . . . . 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))))
251		oveq2 6658	. . . . . . . . . 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)))
252	251	eqeq1d 2624	. . . . . . . . 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)))
253		oveq2 6658	. . . . . . . . . 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)))
254	253	eqeq1d 2624	. . . . . . . . 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)))
255		1m1e0 11089	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
256		f1ofn 6138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
257	45, 256	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
258	257	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
259		imassrn 5477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
260		f1of 6137	. . . . . . . . . . . . . . . . . . . . . 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))
261	212, 260	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
262		frn 6053	. . . . . . . . . . . . . . . . . . . . 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))
263	261, 262	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
264	259, 263	syl5ss 3614	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
265	264	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
266		eqidd 2623	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
267		eluzfz1 12348	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
268	189, 267	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ (1...𝑁))
269	266, 195, 268, 68	fvmptd 6288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
270	269	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
271		f1ofn 6138	. . . . . . . . . . . . . . . . . . . . . 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))
272	212, 271	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
273	272	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
274		fzss2 12381	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
275	236, 274	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
276		eluzfz1 12348	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1)))
277	66, 276	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
278	277	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
279		fnfvima 6496	. . . . . . . . . . . . . . . . . . . 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))))
280	273, 275, 278, 279	syl3anc 1326	. . . . . . . . . . . . . . . . . . 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))))
281	270, 280	eqeltrrd 2702	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
282		fnfvima 6496	. . . . . . . . . . . . . . . . . 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)))))
283	258, 265, 281, 282	syl3anc 1326	. . . . . . . . . . . . . . . . 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)))))
284		imaco 5640	. . . . . . . . . . . . . . . . 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))))
285	283, 284	syl6eleqr 2712	. . . . . . . . . . . . . . . 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))))
286		fnconstg 6093	. . . . . . . . . . . . . . . . . 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))))
287	33, 286	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)))
288		fnconstg 6093	. . . . . . . . . . . . . . . . . 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)))
289	36, 288	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))
290		fvun1 6269	. . . . . . . . . . . . . . . . 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)))
291	287, 289, 290	mp3an12 1414	. . . . . . . . . . . . . . . 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)))
292	226, 285, 291	syl2anc 693	. . . . . . . . . . . . . . 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)))
293	33	fvconst2 6469	. . . . . . . . . . . . . . . 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)
294	285, 293	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)
295	292, 294	eqtrd 2656	. . . . . . . . . . . . . 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)
296	295	oveq1d 6665	. . . . . . . . . . . . 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))
297		fzss1 12380	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
298	66, 297	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
299	298	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
300		eluzfz2 12349	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
301	236, 300	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
302		fnfvima 6496	. . . . . . . . . . . . . . . 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)))
303	258, 299, 301, 302	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
304		fvun2 6270	. . . . . . . . . . . . . . . 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)))
305	35, 38, 304	mp3an12 1414	. . . . . . . . . . . . . . 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)))
306	59, 303, 305	syl2anc 693	. . . . . . . . . . . . . 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)))
307	36	fvconst2 6469	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
308	303, 307	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
309	306, 308	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 0)
310	255, 296, 309	3eqtr4a 2682	. . . . . . . . . . . 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)))
311		fveq2 6191	. . . . . . . . . . . . . 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)))
312	311	oveq1d 6665	. . . . . . . . . . . . 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))
313		fveq2 6191	. . . . . . . . . . . . 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)))
314	312, 313	eqeq12d 2637	. . . . . . . . . . . 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))))
315	310, 314	syl5ibrcom 237	. . . . . . . . . . 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)))
316	315	imp 445	. . . . . . . . . 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))
317	316	adantlr 751	. . . . . . . . 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))
318	249	subid1d 10381	. . . . . . . . . . 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))
319	318	adantr 481	. . . . . . . . . 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))
320		eldifsn 4317	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁)))
321		df-ne 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁) ↔ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁))
322	321	anbi2i 730	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st ‘𝑇))‘𝑁)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)))
323	320, 322	bitri 264	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁)))
324		fnconstg 6093	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
325	36, 324	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))
326	35, 325	pm3.2i 471	. . . . . . . . . . . . . . . 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))))
327		imain 5974	. . . . . . . . . . . . . . . . . 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)))))
328	48, 327	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)))))
329		fzdisj 12368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
330	53, 329	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
331	330	imaeq2d 5466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ((2nd ‘(1st ‘𝑇)) “ ∅))
332	331, 57	syl6eq 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
333	328, 332	sylan9req 2677	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
334		fnun 5997	. . . . . . . . . . . . . . . 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)))))
335	326, 333, 334	sylancr 695	. . . . . . . . . . . . . . 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)))))
336		imaundi 5545	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
337	204, 209	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
338	337	difeq1d 3727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
339		difun2 4048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
340	338, 339	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}))
341		difsn 4328	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
342	164, 341	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
343	340, 342	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
344	343	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
345	73	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑦))
346		fzsplit2 12366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 1) ∈ (ℤ≥‘1) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑦)) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
347	67, 345, 346	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
348	344, 347	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
349	348	imaeq2d 5466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))))
350		imadif 5973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})))
351	48, 350	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})))
352		elfz1end 12371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
353	68, 352	sylib 208	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
354		fnsnfv 6258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
355	257, 353, 354	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
356	355	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {𝑁}) = {((2nd ‘(1st ‘𝑇))‘𝑁)})
357	83, 356	difeq12d 3729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
358	351, 357	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
359	358	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
360	349, 359	eqtr3d 2658	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
361	336, 360	syl5eqr 2670	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
362	361	fneq2d 5982	. . . . . . . . . . . . . . 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)})))
363	335, 362	mpbid 222	. . . . . . . . . . . . . 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)}))
364		incom 3805	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ({((2nd ‘(1st ‘𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
365		disjdif 4040	. . . . . . . . . . . . . . . 16 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)})) = ∅
366	364, 365	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑁) ∖ {((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd ‘(1st ‘𝑇))‘𝑁)}) = ∅
367		fnconstg 6093	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)})
368	33, 367	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)}
369		fvun1 6269	. . . . . . . . . . . . . . . . 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))
370	368, 369	mp3an2 1412	. . . . . . . . . . . . . . . 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))
371		fnconstg 6093	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)})
372	36, 371	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st ‘𝑇))‘𝑁)}
373		fvun1 6269	. . . . . . . . . . . . . . . . 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))
374	372, 373	mp3an2 1412	. . . . . . . . . . . . . . . 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))
375	370, 374	eqtr4d 2659	. . . . . . . . . . . . . . 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))
376	366, 375	mpanr1 719	. . . . . . . . . . . . . 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))
377	363, 376	sylan 488	. . . . . . . . . . . . 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))
378	323, 377	sylan2br 493	. . . . . . . . . . . 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))
379	378	anassrs 680	. . . . . . . . . . 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))
380		imaundi 5545	. . . . . . . . . . . . . . . . . . 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}))
381		imaco 5640	. . . . . . . . . . . . . . . . . . . 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))))
382		imaco 5640	. . . . . . . . . . . . . . . . . . . 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}))
383	381, 382	uneq12i 3765	. . . . . . . . . . . . . . . . . . 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})))
384	380, 383	eqtri 2644	. . . . . . . . . . . . . . . . . 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})))
385		fzpred 12389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ (ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
386	66, 385	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
387		uncom 3757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({1} ∪ ((1 + 1)...(𝑦 + 1))) = (((1 + 1)...(𝑦 + 1)) ∪ {1})
388	386, 387	syl6eq 2672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = (((1 + 1)...(𝑦 + 1)) ∪ {1}))
389	388	imaeq2d 5466	. . . . . . . . . . . . . . . . . . 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})))
390	389	adantl 482	. . . . . . . . . . . . . . . . . 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})))
391		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
392	124	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → ((1 + 1) − 1) = 1)
393		zcn 11382	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
394		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
395	393, 394	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → ((𝑦 + 1) − 1) = 𝑦)
396	392, 395	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = (1...𝑦))
397		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℤ)
398	397	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℂ)
399		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
400	398, 399	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) = 𝑗)
401	400	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
402	401	ibir 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
403	402	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
404		peano2z 11418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
405	404, 117	jctil 560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ℤ → ((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ))
406	397	peano2zd 11485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (𝑗 + 1) ∈ ℤ)
407	406, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
408		fzsubel 12377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
409	405, 407, 408	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
410	403, 409	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → (𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
411	400	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 = ((𝑗 + 1) − 1))
412	411	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → 𝑗 = ((𝑗 + 1) − 1))
413		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = (𝑗 + 1) → (𝑛 − 1) = ((𝑗 + 1) − 1))
414	413	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = (𝑗 + 1) → (𝑗 = (𝑛 − 1) ↔ 𝑗 = ((𝑗 + 1) − 1)))
415	414	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
416	410, 412, 415	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
417	416	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
418		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
419		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ∈ ℤ)
420	419, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
421		fzsubel 12377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
422	405, 420, 421	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
423	418, 422	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
424		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
425	423, 424	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
426	425	rexlimdva 3031	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → (∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
427	417, 426	impbid 202	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
428		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑗 ∈ V
429		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
430	429	elrnmpt 5372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
431	428, 430	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
432	427, 431	syl6bbr 278	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))))
433	432	eqrdv 2620	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
434	396, 433	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
435	391, 434	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
436	435	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
437		df-ima 5127	. . . . . . . . . . . . . . . . . . . . . 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)))
438		uzid 11702	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ≥‘1))
439		peano2uz 11741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ (ℤ≥‘1) → (1 + 1) ∈ (ℤ≥‘1))
440	115, 438, 439	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 1) ∈ (ℤ≥‘1)
441		fzss1 12380	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1 + 1) ∈ (ℤ≥‘1) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1)))
442	440, 441	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1))
443	442, 275	syl5ss 3614	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...𝑁))
444	443	resmptd 5452	. . . . . . . . . . . . . . . . . . . . . . . 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))))
445		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ ((1 + 1)...(𝑦 + 1)) → (1 + 1) ≤ 1)
446	171, 445	mto 188	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ¬ 1 ∈ ((1 + 1)...(𝑦 + 1))
447		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ 1 ∈ ((1 + 1)...(𝑦 + 1))))
448	446, 447	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
449	448	necon2ai 2823	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ≠ 1)
450	449, 182	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
451	450	mpteq2ia 4740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
452	444, 451	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . 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)))
453	452	rneqd 5353	. . . . . . . . . . . . . . . . . . . . . 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)))
454	437, 453	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . 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)))
455	436, 454	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
456	455	imaeq2d 5466	. . . . . . . . . . . . . . . . . . 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)))))
457	269	sneqd 4189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = {𝑁})
458		fnsnfv 6258	. . . . . . . . . . . . . . . . . . . . . . . 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}))
459	272, 268, 458	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
460	457, 459	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑁} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
461	460	imaeq2d 5466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {𝑁}) = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
462	355, 461	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
463	462	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
464	456, 463	uneq12d 3768	. . . . . . . . . . . . . . . . . 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}))))
465	384, 390, 464	3eqtr4a 2682	. . . . . . . . . . . . . . . . 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)}))
466	465	xpeq1d 5138	. . . . . . . . . . . . . . . 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}))
467		xpundir 5172	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}) × {1}) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))
468	466, 467	syl6eq 2672	. . . . . . . . . . . . . . 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})))
469		imaco 5640	. . . . . . . . . . . . . . . . 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)))
470		df-ima 5127	. . . . . . . . . . . . . . . . . . . 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))
471		fzss1 12380	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ≥‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
472	232, 471	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
473	472	resmptd 5452	. . . . . . . . . . . . . . . . . . . . . 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))))
474		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ ℝ)
475	64	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
476	475	peano2zd 11485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℤ)
477	476	zred 11482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
478	64	nnge1d 11063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ≤ (𝑦 + 1))
479	474, 219, 477, 478, 220	lelttrd 10195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 < ((𝑦 + 1) + 1))
480	474, 477	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1 < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ 1))
481	479, 480	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ ((𝑦 + 1) + 1) ≤ 1)
482		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ 1)
483	481, 482	nsyl 135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁))
484		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 1 → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
485	484	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 1 → (¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
486	483, 485	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 = 1 → ¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)))
487	486	necon2ad 2809	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ≠ 1))
488	487	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ≠ 1)
489	488, 182	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
490	489	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
491	490	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
492	473, 491	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 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)))
493	492	rneqd 5353	. . . . . . . . . . . . . . . . . . . 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)))
494	470, 493	syl5eq 2668	. . . . . . . . . . . . . . . . . . 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)))
495		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))
496	495	elrnmpt 5372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
497	428, 496	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
498		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁))
499	114, 476	anim12ci 591	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
500		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ∈ ℤ)
501	500, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
502		fzsubel 12377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
503	499, 501, 502	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
504	498, 503	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
505		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
506	504, 505	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
507	506	rexlimdva 3031	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
508		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℤ)
509	508	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℂ)
510	509, 399	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) = 𝑗)
511	510	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
512	511	ibir 257	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
513	512	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
514	508	peano2zd 11485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (𝑗 + 1) ∈ ℤ)
515	514, 115	jctir 561	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
516		fzsubel 12377	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
517	499, 515, 516	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . 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))))
518	513, 517	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → (𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
519	510	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 = ((𝑗 + 1) − 1))
520	519	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → 𝑗 = ((𝑗 + 1) − 1))
521	414	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
522	518, 520, 521	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
523	522	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
524	507, 523	impbid 202	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
525	497, 524	syl5bb 272	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
526	525	eqrdv 2620	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
527	64	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℂ)
528		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) ∈ ℂ → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
529	527, 528	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
530	529	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
531	530	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
532	494, 526, 531	3eqtrd 2660	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ((𝑦 + 1)...(𝑁 − 1)))
533	532	imaeq2d 5466	. . . . . . . . . . . . . . . . 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))))
534	469, 533	syl5eq 2668	. . . . . . . . . . . . . . . 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))))
535	534	xpeq1d 5138	. . . . . . . . . . . . . . 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}))
536	468, 535	uneq12d 3768	. . . . . . . . . . . . . 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})))
537		un23 3772	. . . . . . . . . . . . . 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}))
538	536, 537	syl6eq 2672	. . . . . . . . . . . . 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})))
539	538	fveq1d 6193	. . . . . . . . . . . 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))
540	539	ad2antrr 762	. . . . . . . . . . 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))
541		imaundi 5545	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
542		fzsplit2 12366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
543	234, 202, 542	syl2anr 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
544	208	uneq2d 3767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
545	544	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
546	543, 545	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
547	546	imaeq2d 5466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})))
548	355	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st ‘𝑇)) “ {𝑁}))
549	548	uneq2d 3767	. . . . . . . . . . . . . . . . . 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})))
550	541, 547, 549	3eqtr4a 2682	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st ‘𝑇))‘𝑁)}))
551	550	xpeq1d 5138	. . . . . . . . . . . . . . . 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}))
552		xpundir 5172	. . . . . . . . . . . . . . . 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}))
553	551, 552	syl6eq 2672	. . . . . . . . . . . . . . 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})))
554	553	uneq2d 3767	. . . . . . . . . . . . . 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}))))
555		unass 3770	. . . . . . . . . . . . . 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})))
556	554, 555	syl6eqr 2674	. . . . . . . . . . . . 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})))
557	556	fveq1d 6193	. . . . . . . . . . . 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))
558	557	ad2antrr 762	. . . . . . . . . . 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))
559	379, 540, 558	3eqtr4d 2666	. . . . . . . . . 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))
560	319, 559	eqtrd 2656	. . . . . . . . 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))
561	252, 254, 317, 560	ifbothda 4123	. . . . . . . 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))
562	561	oveq2d 6666	. . . . . . 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)))
563	250, 562	eqtr2d 2657	. . . . . 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)))
564	563	mpteq2dva 4744	. . . . 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))))
565	93, 564	eqtrd 2656	. . . 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))))
566	52	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
567	160	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
568	158	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
569		elfzle2 12345	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
570	569	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
571	159	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
572	566, 567, 568, 570, 571	lelttrd 10195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
573		poimirlem21.4	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)
574	573	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑇) = 𝑁)
575	572, 574	breqtrrd 4681	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd ‘𝑇))
576	575	iftrued 4094	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
577	576	csbeq1d 3540	. . . . 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}))))
578		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
579		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
580	579	imaeq2d 5466	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑦)))
581	580	xpeq1d 5138	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}))
582		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
583	582	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
584	583	imaeq2d 5466	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
585	584	xpeq1d 5138	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
586	581, 585	uneq12d 3768	. . . . . . 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})))
587	586	oveq2d 6666	. . . . . 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}))))
588	578, 587	csbie 3559	. . . . 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})))
589	577, 588	syl6eq 2672	. . . 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}))))
590		ovexd 6680	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0)) ∈ V)
591		fvexd 6203	. . . . 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)
592		eqidd 2623	. . . . 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))))
593		ffn 6045	. . . . . . 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))
594	247, 593	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))
595		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(2nd ‘(1st ‘𝑇))
596		nfmpt1 4747	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
597	595, 596	nfco 5287	. . . . . . . . . 10 ⊢ Ⅎ𝑛((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
598		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1...(𝑦 + 1))
599	597, 598	nfima 5474	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
600		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑛{1}
601	599, 600	nfxp 5142	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})
602		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑛(((𝑦 + 1) + 1)...𝑁)
603	597, 602	nfima 5474	. . . . . . . . 9 ⊢ Ⅎ𝑛(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
604		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑛{0}
605	603, 604	nfxp 5142	. . . . . . . 8 ⊢ Ⅎ𝑛((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})
606	601, 605	nfun 3769	. . . . . . 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}))
607	606	dffn5f 6252	. . . . . 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)))
608	594, 607	sylib 208	. . . . 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)))
609	89, 590, 591, 592, 608	offval2 6914	. . . 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))))
610	565, 589, 609	3eqtr4rd 2667	. . 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}))))
611	610	mpteq2dva 4744	. 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})))))
612	22, 611	eqtr4d 2659	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3200  ⦋csb 3533   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466

This theorem is referenced by:  poimirlem20  33429