poimirlem7 - Mathbox for Brendan Leahy

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Theorem poimirlem7 38009

Description: Lemma for poimir 38035, similar to poimirlem6 38008, but for vertices after the opposite vertex. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem9.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem9.2	⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
poimirlem7.3	⊢ (𝜑 → 𝑀 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))

**Assertion**
Ref	Expression
poimirlem7	⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2^nd ‘(1^st ‘𝑇))‘𝑀))

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

**Proof of Theorem poimirlem7**
Step	Hyp	Ref	Expression
1		poimirlem9.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		elrabi 3627	. . . . . . . . 9 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3		poimirlem22.s	. . . . . . . . 9 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4	2, 3	eleq2s 2859	. . . . . . . 8 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6		xp1st 7967	. . . . . . 7 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8		xp2nd 7968	. . . . . 6 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
10		fvex 6844	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
11		f1oeq1 6759	. . . . . 6 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
12	10, 11	elab 3619	. . . . 5 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
13	9, 12	sylib 220	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14		f1of 6771	. . . 4 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
16		poimirlem9.2	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
17		elfznn 13502	. . . . . . . . 9 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
19	18	peano2nnd 12186	. . . . . . 7 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℕ)
20	19	peano2nnd 12186	. . . . . 6 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ∈ ℕ)
21		nnuz 12822	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2851	. . . . 5 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ∈ (ℤ_≥‘1))
23		fzss1 13512	. . . . 5 ⊢ ((((2^nd ‘𝑇) + 1) + 1) ∈ (ℤ_≥‘1) → ((((2^nd ‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁))
24	22, 23	syl 17	. . . 4 ⊢ (𝜑 → ((((2^nd ‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁))
25		poimirlem7.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁))
26	24, 25	sseldd 3918	. . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
27	15, 26	ffvelcdmd 7030	. 2 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁))
28		xp1st 7967	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
29	7, 28	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
30		elmapfn 8806	. . . . . . . . . . . 12 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
32	31	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
33		1ex 11135	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
34		fnconstg 6719	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))))
35	33, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1)))
36		c0ex 11133	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
37		fnconstg 6719	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
38	36, 37	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))
39	35, 38	pm3.2i 472	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
40		dff1o3 6777	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑇))))
41	40	simprbi 499	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
42	13, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
43		imain 6574	. . . . . . . . . . . . . . 15 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
44	42, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
45	25	elfzelzd 13474	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	45	zred 12628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℝ)
47	46	ltm1d 12083	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
48		fzdisj 13500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
50	49	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
51		ima0 6036	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
52	50, 51	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
53	44, 52	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = ∅)
54		fnun 6603	. . . . . . . . . . . . 13 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
55	39, 53, 54	sylancr 594	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
56	45	zcnd 12629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℂ)
57		npcan1 11570	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
59		1red 11140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 1 ∈ ℝ)
60	20	nnred 12184	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ∈ ℝ)
61	19	nnred 12184	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ ℝ)
62	19	nnge1d 12220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ≤ ((2^nd ‘𝑇) + 1))
63	61	ltp1d 12081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) < (((2^nd ‘𝑇) + 1) + 1))
64	59, 61, 60, 62, 63	lelttrd 11299	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 < (((2^nd ‘𝑇) + 1) + 1))
65		elfzle1 13476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → (((2^nd ‘𝑇) + 1) + 1) ≤ 𝑀)
66	25, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) ≤ 𝑀)
67	59, 60, 46, 64, 66	ltletrd 11301	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 1 < 𝑀)
68	59, 46, 67	ltled 11289	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ≤ 𝑀)
69		elnnz1 12548	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
70	45, 68, 69	sylanbrc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℕ)
71	70, 21	eleqtrdi 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
72	58, 71	eqeltrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘1))
73		peano2zm 12565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
74	45, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
75		uzid 12798	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
76		peano2uz 12846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
77	74, 75, 76	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
78	58, 77	eqeltrrd 2842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑀 − 1)))
79		uzss 12806	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘(𝑀 − 1)) → (ℤ_≥‘𝑀) ⊆ (ℤ_≥‘(𝑀 − 1)))
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ_≥‘𝑀) ⊆ (ℤ_≥‘(𝑀 − 1)))
81		elfzuz3 13470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
82	25, 81	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
83	80, 82	sseldd 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 − 1)))
84		fzsplit2 13498	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑀 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
85	72, 83, 84	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
86	58	oveq1d 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
87	86	uneq2d 4101	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
88	85, 87	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
89	88	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
90		imaundi 6104	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
91	89, 90	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
92		f1ofo 6778	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
93	13, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
94		foima 6748	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
95	93, 94	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
96	91, 95	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
97	96	fneq2d 6583	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)))
98	55, 97	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
99	98	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
100		ovexd 7395	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → (1...𝑁) ∈ V)
101		inidm 4158	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
102		eqidd 2742	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
103		imaundi 6104	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ {𝑀}) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
104		fzpred 13521	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
105	82, 104	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
106	105	imaeq2d 6019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
107		f1ofn 6772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
108	13, 107	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
109		fnsnfv 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑇))‘𝑀)} = ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
110	108, 26, 109	syl2anc 591	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘𝑀)} = ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
111	110	uneq1d 4100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ {𝑀}) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
112	103, 106, 111	3eqtr4a 2802	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) = ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
113	112	xpeq1d 5650	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}))
114		xpundir 5691	. . . . . . . . . . . . . . . 16 ⊢ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
115	113, 114	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
116	115	uneq2d 4101	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
117		un12 4105	. . . . . . . . . . . . . 14 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
118	116, 117	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
119	118	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
120	119	ad2antrr 733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
121		fnconstg 6719	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
122	36, 121	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))
123	35, 122	pm3.2i 472	. . . . . . . . . . . . . . 15 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
124		imain 6574	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
125	42, 124	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
126	74	zred 12628	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
127		peano2re 11314	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
128	46, 127	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
129	46	ltp1d 12081	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
130	126, 46, 128, 47, 129	lttrd 11302	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 − 1) < (𝑀 + 1))
131		fzdisj 13500	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
132	130, 131	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
133	132	imaeq2d 6019	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
134	133, 51	eqtrdi 2792	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
135	125, 134	eqtr3d 2778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
136		fnun 6603	. . . . . . . . . . . . . . 15 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
137	123, 135, 136	sylancr 594	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
138		imaundi 6104	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
139		imadif 6573	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀})))
140	42, 139	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀})))
141		fzsplit 13499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
142	26, 141	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
143	142	difeq1d 4059	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
144		difundir 4222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
145		fzsplit2 13498	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑀 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
146	72, 78, 145	syl2anc 591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
147	58	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
148		fzsn 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
149	45, 148	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
150	147, 149	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
151	150	uneq2d 4101	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
152	146, 151	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
153	152	difeq1d 4059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
154		difun2 4412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
155	126, 46	ltnled 11288	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
156	47, 155	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
157		elfzle2 13477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
158	156, 157	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
159		difsn 4734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
160	158, 159	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
161	154, 160	eqtrid 2788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
162	153, 161	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
163	46, 128	ltnled 11288	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
164	129, 163	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
165		elfzle1 13476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
166	164, 165	nsyl 140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
167		difsn 4734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
168	166, 167	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
169	162, 168	uneq12d 4102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
170	144, 169	eqtrid 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
171	143, 170	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
172	171	imaeq2d 6019	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
173	110	eqcomd 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}) = {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
174	95, 173	difeq12d 4061	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
175	140, 172, 174	3eqtr3d 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
176	138, 175	eqtr3id 2790	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
177	176	fneq2d 6583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})))
178	137, 177	mpbid 234	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
179		eldifsn 4722	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
180	179	biimpri 230	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
181	180	ancoms 460	. . . . . . . . . . . . 13 ⊢ ((𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
182		disjdif 4403	. . . . . . . . . . . . . 14 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅
183		fnconstg 6719	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
184	36, 183	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)}
185		fvun2 6923	. . . . . . . . . . . . . . 15 ⊢ ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
186	184, 185	mp3an1 1457	. . . . . . . . . . . . . 14 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
187	182, 186	mpanr1 710	. . . . . . . . . . . . 13 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
188	178, 181, 187	syl2an 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
189	188	anassrs 469	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
190	120, 189	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
191	32, 99, 100, 100, 101, 102, 190	ofval 7635	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)))
192		fnconstg 6719	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
193	33, 192	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))
194	193, 122	pm3.2i 472	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
195		imain 6574	. . . . . . . . . . . . . . 15 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
196	42, 195	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
197		fzdisj 13500	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
198	129, 197	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
199	198	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
200	199, 51	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
201	196, 200	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
202		fnun 6603	. . . . . . . . . . . . 13 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
203	194, 201, 202	sylancr 594	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
204	142	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
205		imaundi 6104	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
206	204, 205	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
207	206, 95	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
208	207	fneq2d 6583	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
209	203, 208	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
210	209	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
211		imaundi 6104	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
212	152	imaeq2d 6019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
213	110	uneq2d 4101	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀})))
214	211, 212, 213	3eqtr4a 2802	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
215	214	xpeq1d 5650	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) × {1}))
216		xpundir 5691	. . . . . . . . . . . . . . . 16 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}))
217	215, 216	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1})))
218	217	uneq1d 4100	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
219		un23 4106	. . . . . . . . . . . . . . 15 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}))
220	219	equncomi 4093	. . . . . . . . . . . . . 14 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1})) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
221	218, 220	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
222	221	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
223	222	ad2antrr 733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
224		fnconstg 6719	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
225	33, 224	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)}
226		fvun2 6923	. . . . . . . . . . . . . . 15 ⊢ ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
227	225, 226	mp3an1 1457	. . . . . . . . . . . . . 14 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ (({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
228	182, 227	mpanr1 710	. . . . . . . . . . . . 13 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
229	178, 181, 228	syl2an 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
230	229	anassrs 469	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
231	223, 230	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))
232	32, 210, 100, 100, 101, 102, 231	ofval 7635	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)))
233	191, 232	eqtr4d 2779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
234	233	an32s 659	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
235	234	anasss 468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
236		fveq2 6831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
237	236	breq2d 5087	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
238	237	ifbid 4481	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
239	238	csbeq1d 3837	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
240		2fveq3 6836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
241		2fveq3 6836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
242	241	imaeq1d 6018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
243	242	xpeq1d 5650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
244	241	imaeq1d 6018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
245	244	xpeq1d 5650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
246	243, 245	uneq12d 4102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
247	240, 246	oveq12d 7378	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
248	247	csbeq2dv 3840	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
249	239, 248	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
250	249	mpteq2dv 5169	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
251	250	eqeq2d 2752	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
252	251, 3	elrab2 3634	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
253	252	simprbi 499	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
254	1, 253	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
255		breq1 5078	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀 − 2) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 2) < (2^nd ‘𝑇)))
256	255	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 2) < (2^nd ‘𝑇)))
257		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀 − 2) → (𝑦 + 1) = ((𝑀 − 2) + 1))
258		sub1m1 12424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) − 1) = (𝑀 − 2))
259	56, 258	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 − 1) − 1) = (𝑀 − 2))
260	259	oveq1d 7375	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = ((𝑀 − 2) + 1))
261	74	zcnd 12629	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
262		npcan1 11570	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 − 1) ∈ ℂ → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
263	261, 262	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1))
264	260, 263	eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 − 2) + 1) = (𝑀 − 1))
265	257, 264	sylan9eqr 2798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 + 1) = (𝑀 − 1))
266	256, 265	ifbieq2d 4484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 2) < (2^nd ‘𝑇), 𝑦, (𝑀 − 1)))
267	18	nncnd 12185	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℂ)
268		add1p1 12423	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑇) ∈ ℂ → (((2^nd ‘𝑇) + 1) + 1) = ((2^nd ‘𝑇) + 2))
269	267, 268	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) + 1) = ((2^nd ‘𝑇) + 2))
270	269, 66	eqbrtrrd 5099	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) + 2) ≤ 𝑀)
271	18	nnred 12184	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
272		2re 12250	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
273		leaddsub 11621	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2^nd ‘𝑇) + 2) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 2)))
274	272, 273	mp3an2 1458	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2^nd ‘𝑇) + 2) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 2)))
275	271, 46, 274	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) + 2) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 2)))
276	59, 46	posdifd 11732	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 < 𝑀 ↔ 0 < (𝑀 − 1)))
277	67, 276	mpbid 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 < (𝑀 − 1))
278		elnnz 12529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 − 1) ∈ ℕ ↔ ((𝑀 − 1) ∈ ℤ ∧ 0 < (𝑀 − 1)))
279	74, 277, 278	sylanbrc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ)
280		nnm1nn0 12473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 − 1) ∈ ℕ → ((𝑀 − 1) − 1) ∈ ℕ₀)
281	279, 280	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 − 1) − 1) ∈ ℕ₀)
282	259, 281	eqeltrrd 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 2) ∈ ℕ₀)
283	282	nn0red 12494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 2) ∈ ℝ)
284	271, 283	lenltd 11287	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) ≤ (𝑀 − 2) ↔ ¬ (𝑀 − 2) < (2^nd ‘𝑇)))
285	275, 284	bitrd 281	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) + 2) ≤ 𝑀 ↔ ¬ (𝑀 − 2) < (2^nd ‘𝑇)))
286	270, 285	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (𝑀 − 2) < (2^nd ‘𝑇))
287	286	iffalsed 4468	. . . . . . . . . . . . 13 ⊢ (𝜑 → if((𝑀 − 2) < (2^nd ‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
288	287	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if((𝑀 − 2) < (2^nd ‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1))
289	266, 288	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
290	289	csbeq1d 3837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
291		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
292	291	imaeq2d 6019	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑀 − 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))))
293	292	xpeq1d 5650	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑀 − 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}))
294	293	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}))
295		oveq1 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
296	295, 58	sylan9eqr 2798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
297	296	oveq1d 7375	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
298	297	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
299	298	xpeq1d 5650	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))
300	294, 299	uneq12d 4102	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))
301	300	oveq2d 7376	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
302	74, 301	csbied 3869	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
303	302	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋(𝑀 − 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
304	290, 303	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
305		poimir.0	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
306		nnm1nn0 12473	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
307	305, 306	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
308	305	nnred 12184	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
309	46	lem1d 12084	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 − 1) ≤ 𝑀)
310		elfzle2 13477	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ((((2^nd ‘𝑇) + 1) + 1)...𝑁) → 𝑀 ≤ 𝑁)
311	25, 310	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
312	126, 46, 308, 309, 311	letrd 11298	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 1) ≤ 𝑁)
313	126, 308, 59, 312	lesub1dd 11761	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 − 1) − 1) ≤ (𝑁 − 1))
314	259, 313	eqbrtrrd 5099	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 2) ≤ (𝑁 − 1))
315		elfz2nn0 13567	. . . . . . . . . 10 ⊢ ((𝑀 − 2) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 2) ∈ ℕ₀ ∧ (𝑁 − 1) ∈ ℕ₀ ∧ (𝑀 − 2) ≤ (𝑁 − 1)))
316	282, 307, 314, 315	syl3anbrc 1351	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 2) ∈ (0...(𝑁 − 1)))
317		ovexd 7395	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V)
318	254, 304, 316, 317	fvmptd 6947	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑀 − 2)) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
319	318	fveq1d 6833	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
320	319	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
321		breq1 5078	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < (2^nd ‘𝑇)))
322	321	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < (2^nd ‘𝑇)))
323		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1))
324	323, 58	sylan9eqr 2798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀)
325	322, 324	ifbieq2d 4484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2^nd ‘𝑇), 𝑦, 𝑀))
326	61	lep1d 12082	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ≤ (((2^nd ‘𝑇) + 1) + 1))
327	61, 60, 46, 326, 66	letrd 11298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ≤ 𝑀)
328		1re 11139	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
329		leaddsub 11621	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2^nd ‘𝑇) + 1) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 1)))
330	328, 329	mp3an2 1458	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2^nd ‘𝑇) + 1) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 1)))
331	271, 46, 330	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) ≤ 𝑀 ↔ (2^nd ‘𝑇) ≤ (𝑀 − 1)))
332	271, 126	lenltd 11287	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘𝑇) ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < (2^nd ‘𝑇)))
333	331, 332	bitrd 281	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) ≤ 𝑀 ↔ ¬ (𝑀 − 1) < (2^nd ‘𝑇)))
334	327, 333	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (𝑀 − 1) < (2^nd ‘𝑇))
335	334	iffalsed 4468	. . . . . . . . . . . . 13 ⊢ (𝜑 → if((𝑀 − 1) < (2^nd ‘𝑇), 𝑦, 𝑀) = 𝑀)
336	335	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < (2^nd ‘𝑇), 𝑦, 𝑀) = 𝑀)
337	325, 336	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
338	337	csbeq1d 3837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
339		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
340	339	imaeq2d 6019	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
341	340	xpeq1d 5650	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}))
342		oveq1 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
343	342	oveq1d 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
344	343	imaeq2d 6019	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
345	344	xpeq1d 5650	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
346	341, 345	uneq12d 4102	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
347	346	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑀 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
348	347	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
349	25, 348	csbied 3869	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
350	349	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
351	338, 350	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
352	279	nnnn0d 12493	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
353	46, 308, 59, 311	lesub1dd 11761	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
354		elfz2nn0 13567	. . . . . . . . . 10 ⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ₀ ∧ (𝑁 − 1) ∈ ℕ₀ ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
355	352, 307, 353, 354	syl3anbrc 1351	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
356		ovexd 7395	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
357	254, 351, 355, 356	fvmptd 6947	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
358	357	fveq1d 6833	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
359	358	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛))
360	235, 320, 359	3eqtr4d 2786	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛))
361	360	expr 458	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛)))
362	361	necon1d 2958	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) → 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
363		elmapi 8790	. . . . . . . . . . 11 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
364	29, 363	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
365	364, 27	ffvelcdmd 7030	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ (0..^𝐾))
366		elfzonn0 13657	. . . . . . . . 9 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℕ₀)
367	365, 366	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℕ₀)
368	367	nn0red 12494	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℝ)
369	368	ltp1d 12081	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) < (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
370	368, 369	ltned 11277	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
371	318	fveq1d 6833	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
372		ovexd 7395	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ V)
373		eqidd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
374		fzss1 13512	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
375	71, 374	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
376		eluzfz1 13480	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
377	82, 376	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
378		fnfvima 7181	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
379	108, 375, 377, 378	syl3anc 1380	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
380		fvun2 6923	. . . . . . . . . . . . 13 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
381	35, 38, 380	mp3an12 1460	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
382	53, 379, 381	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
383	36	fvconst2 7152	. . . . . . . . . . . 12 ⊢ (((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) → ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 0)
384	379, 383	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 0)
385	382, 384	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 0)
386	385	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 0)
387	31, 98, 372, 372, 101, 373, 386	ofval 7635	. . . . . . . 8 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 0))
388	27, 387	mpdan 694	. . . . . . 7 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 0))
389	367	nn0cnd 12495	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℂ)
390	389	addridd 11341	. . . . . . 7 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 0) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
391	371, 388, 390	3eqtrd 2780	. . . . . 6 ⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
392	357	fveq1d 6833	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
393		fzss2 13513	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (1...𝑀) ⊆ (1...𝑁))
394	82, 393	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ⊆ (1...𝑁))
395		elfz1end 13503	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
396	70, 395	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
397		fnfvima 7181	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
398	108, 394, 396, 397	syl3anc 1380	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
399		fvun1 6922	. . . . . . . . . . . . 13 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
400	193, 122, 399	mp3an12 1460	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
401	201, 398, 400	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
402	33	fvconst2 7152	. . . . . . . . . . . 12 ⊢ (((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 1)
403	398, 402	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 1)
404	401, 403	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 1)
405	404	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 1)
406	31, 209, 372, 372, 101, 373, 405	ofval 7635	. . . . . . . 8 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
407	27, 406	mpdan 694	. . . . . . 7 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
408	392, 407	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
409	370, 391, 408	3netr4d 3013	. . . . 5 ⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
410		fveq2 6831	. . . . . 6 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 2))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
411		fveq2 6831	. . . . . 6 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
412	410, 411	neeq12d 2997	. . . . 5 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ ((𝐹‘(𝑀 − 2))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀))))
413	409, 412	syl5ibrcom 249	. . . 4 ⊢ (𝜑 → (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
414	413	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)))
415	362, 414	impbid 214	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
416	27, 415	riota5 7346	1 ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2^nd ‘(1^st ‘𝑇))‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 {cab 2719 ≠ wne 2936 {crab 3393 Vcvv 3433 ⦋csb 3833 ∖ cdif 3882 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ∅c0 4264 ifcif 4457 {csn 4558 class class class wbr 5075 ↦ cmpt 5156 × cxp 5619 ^◡ccnv 5620 “ cima 5624 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 –onto→wfo 6487 –1-1-onto→wf1o 6488 ‘cfv 6489 ℩crio 7316 (class class class)co 7360 ∘_f cof 7622 1^st c1st 7933 2^nd c2nd 7934 ↑_m cmap 8767 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 < clt 11174 ≤ cle 11175 − cmin 11372 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ...cfz 13456 ..^cfzo 13603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604

This theorem is referenced by: poimirlem8 38010

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