poimirlem6 - Mathbox for Brendan Leahy

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Theorem poimirlem6 32585

Description: Lemma for poimir 32612 establishing, for a face of a simplex defined by a walk along the edges of an 𝑁-cube, the single dimension in which successive vertices before the opposite vertex differ. (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)))
poimirlem6.3	⊢ (𝜑 → 𝑀 ∈ (1...((2^nd ‘𝑇) − 1)))

**Assertion**
Ref	Expression
poimirlem6	⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘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 poimirlem6**
Step	Hyp	Ref	Expression
1		poimirlem9.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		elrabi 3328	. . . . . . . . 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 2706	. . . . . . . 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 7089	. . . . . . 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 7090	. . . . . 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 6113	. . . . . 6 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
11		f1oeq1 6040	. . . . . 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 3319	. . . . 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 207	. . . 4 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
14		f1of 6050	. . . 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 12241	. . . . . . . . 9 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
19	18	nnzd 11357	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℤ)
20		peano2zm 11297	. . . . . . 7 ⊢ ((2^nd ‘𝑇) ∈ ℤ → ((2^nd ‘𝑇) − 1) ∈ ℤ)
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℤ)
22		poimir.0	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
23	22	nnzd 11357	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	21	zred 11358	. . . . . . 7 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ∈ ℝ)
25	18	nnred 10912	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
26	22	nnred 10912	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
27	25	lem1d 10836	. . . . . . 7 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ≤ (2^nd ‘𝑇))
28		nnm1nn0 11211	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
29	22, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
30	29	nn0red 11229	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
31		elfzle2 12216	. . . . . . . . 9 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ≤ (𝑁 − 1))
32	16, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘𝑇) ≤ (𝑁 − 1))
33	26	lem1d 10836	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁)
34	25, 30, 26, 32, 33	letrd 10073	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘𝑇) ≤ 𝑁)
35	24, 25, 26, 27, 34	letrd 10073	. . . . . 6 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ≤ 𝑁)
36		eluz2 11569	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) ↔ (((2^nd ‘𝑇) − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((2^nd ‘𝑇) − 1) ≤ 𝑁))
37	21, 23, 35, 36	syl3anbrc 1239	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
38		fzss2 12252	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) → (1...((2^nd ‘𝑇) − 1)) ⊆ (1...𝑁))
39	37, 38	syl 17	. . . 4 ⊢ (𝜑 → (1...((2^nd ‘𝑇) − 1)) ⊆ (1...𝑁))
40		poimirlem6.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...((2^nd ‘𝑇) − 1)))
41	39, 40	sseldd 3569	. . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
42	15, 41	ffvelrnd 6268	. 2 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ (1...𝑁))
43		xp1st 7089	. . . . . . . . . . . . 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)))
44	7, 43	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)))
45		elmapfn 7766	. . . . . . . . . . . 12 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
48		1ex 9914	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
49		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))))
50	48, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1)))
51		c0ex 9913	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
52		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
53	51, 52	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))
54	50, 53	pm3.2i 470	. . . . . . . . . . . . 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)))
55		dff1o3 6056	. . . . . . . . . . . . . . . . 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))))
56	55	simprbi 479	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
57	13, 56	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
58		imain 5888	. . . . . . . . . . . . . . 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))))
59	57, 58	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))))
60		elfznn 12241	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...((2^nd ‘𝑇) − 1)) → 𝑀 ∈ ℕ)
61	40, 60	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℕ)
62	61	nnred 10912	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ltm1d 10835	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
64		fzdisj 12239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
65	63, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
66	65	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
67		ima0 5400	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
68	66, 67	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
69	59, 68	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = ∅)
70		fnun 5911	. . . . . . . . . . . . 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))))
71	54, 69, 70	sylancr 694	. . . . . . . . . . . 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))))
72	61	nncnd 10913	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℂ)
73		npcan1 10334	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
75		nnuz 11599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ = (ℤ_≥‘1)
76	61, 75	syl6eleq 2698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
77	74, 76	eqeltrd 2688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘1))
78		nnm1nn0 11211	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
79	61, 78	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
80	79	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
81		uzid 11578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
83		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
85	74, 84	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑀 − 1)))
86		uzss 11584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘(𝑀 − 1)) → (ℤ_≥‘𝑀) ⊆ (ℤ_≥‘(𝑀 − 1)))
87	85, 86	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ_≥‘𝑀) ⊆ (ℤ_≥‘(𝑀 − 1)))
88	61	nnzd 11357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℤ)
89		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ (1...((2^nd ‘𝑇) − 1)) → 𝑀 ≤ ((2^nd ‘𝑇) − 1))
90	40, 89	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ≤ ((2^nd ‘𝑇) − 1))
91	62, 24, 26, 90, 35	letrd 10073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
92		eluz2 11569	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
93	88, 23, 91, 92	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
94	87, 93	sseldd 3569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 − 1)))
95		fzsplit2 12237	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑀 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
96	77, 94, 95	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
97	74	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
98	97	uneq2d 3729	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
99	96, 98	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
100	99	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
101		imaundi 5464	. . . . . . . . . . . . . . 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)))
102	100, 101	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))))
103		f1ofo 6057	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
104	13, 103	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
105		foima 6033	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
106	104, 105	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
107	102, 106	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁))) = (1...𝑁))
108	107	fneq2d 5896	. . . . . . . . . . . 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)))
109	71, 108	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
110	109	adantr 480	. . . . . . . . . 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))
111		ovex 6577	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
112	111	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → (1...𝑁) ∈ V)
113		inidm 3784	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
114		eqidd 2611	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
115		imaundi 5464	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ {𝑀}) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
116		fzpred 12259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
117	93, 116	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
118	117	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))))
119		f1ofn 6051	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
120	13, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
121		fnsnfv 6168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑇))‘𝑀)} = ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
122	120, 41, 121	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘𝑀)} = ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
123	122	uneq1d 3728	. . . . . . . . . . . . . . . . . 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))))
124	115, 118, 123	3eqtr4a 2670	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) = ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
125	124	xpeq1d 5062	. . . . . . . . . . . . . . . 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}))
126		xpundir 5095	. . . . . . . . . . . . . . . 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}))
127	125, 126	syl6eq 2660	. . . . . . . . . . . . . . 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})))
128	127	uneq2d 3729	. . . . . . . . . . . . . 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}))))
129		un12 3733	. . . . . . . . . . . . . 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})))
130	128, 129	syl6eq 2660	. . . . . . . . . . . . 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}))))
131	130	fveq1d 6105	. . . . . . . . . . . 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))
132	131	ad2antrr 758	. . . . . . . . . . 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))
133		fnconstg 6006	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
134	51, 133	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))
135	50, 134	pm3.2i 470	. . . . . . . . . . . . . . 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)))
136		imain 5888	. . . . . . . . . . . . . . . . 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))))
137	57, 136	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))))
138	79	nn0red 11229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
139		peano2re 10088	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
140	62, 139	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
141	62	ltp1d 10833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
142	138, 62, 140, 63, 141	lttrd 10077	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 − 1) < (𝑀 + 1))
143		fzdisj 12239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
144	142, 143	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅)
145	144	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
146	145, 67	syl6eq 2660	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅)
147	137, 146	eqtr3d 2646	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
148		fnun 5911	. . . . . . . . . . . . . . 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))))
149	135, 147, 148	sylancr 694	. . . . . . . . . . . . . 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))))
150		imaundi 5464	. . . . . . . . . . . . . . . 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)))
151		imadif 5887	. . . . . . . . . . . . . . . . . 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})))
152	57, 151	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})))
153		fzsplit 12238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
154	41, 153	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
155	154	difeq1d 3689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}))
156		difundir 3839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀}))
157		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑀 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
158	77, 85, 157	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)))
159	74	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀))
160		fzsn 12254	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
161	88, 160	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
162	159, 161	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀})
163	162	uneq2d 3729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀}))
164	158, 163	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀}))
165	164	difeq1d 3689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}))
166		difun2 4000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀})
167	138, 62	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1)))
168	63, 167	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1))
169		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1))
170	168, 169	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1)))
171		difsn 4269	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
172	170, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1)))
173	166, 172	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1)))
174	165, 173	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1)))
175	62, 140	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
176	141, 175	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
177		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀)
178	176, 177	nsyl 134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁))
179		difsn 4269	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
180	178, 179	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
181	174, 180	uneq12d 3730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
182	156, 181	syl5eq 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
183	155, 182	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))
184	183	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))))
185	122	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}) = {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
186	106, 185	difeq12d 3691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
187	152, 184, 186	3eqtr3d 2652	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
188	150, 187	syl5eqr 2658	. . . . . . . . . . . . . . 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)}))
189	188	fneq2d 5896	. . . . . . . . . . . . . 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)})))
190	149, 189	mpbid 221	. . . . . . . . . . . . 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)}))
191		eldifsn 4260	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
192	191	biimpri 217	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
193	192	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
194		disjdif 3992	. . . . . . . . . . . . . 14 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^nd ‘(1^st ‘𝑇))‘𝑀)})) = ∅
195		fnconstg 6006	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
196	51, 195	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {0}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)}
197		fvun2 6180	. . . . . . . . . . . . . . 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))
198	196, 197	mp3an1 1403	. . . . . . . . . . . . . 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))
199	194, 198	mpanr1 715	. . . . . . . . . . . . 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))
200	190, 193, 199	syl2an 493	. . . . . . . . . . . 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))
201	200	anassrs 678	. . . . . . . . . . 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))
202	132, 201	eqtrd 2644	. . . . . . . . . 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))
203	47, 110, 112, 112, 113, 114, 202	ofval 6804	. . . . . . . . 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)))
204		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
205	48, 204	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))
206	205, 134	pm3.2i 470	. . . . . . . . . . . . 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)))
207		imain 5888	. . . . . . . . . . . . . . 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))))
208	57, 207	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))))
209		fzdisj 12239	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
210	141, 209	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
211	210	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
212	211, 67	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
213	208, 212	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
214		fnun 5911	. . . . . . . . . . . . 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))))
215	206, 213, 214	sylancr 694	. . . . . . . . . . . 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))))
216	154	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
217		imaundi 5464	. . . . . . . . . . . . . . 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)))
218	216, 217	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
219	218, 106	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
220	219	fneq2d 5896	. . . . . . . . . . . 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)))
221	215, 220	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
222	221	adantr 480	. . . . . . . . . 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))
223		imaundi 5464	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ {𝑀}))
224	164	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})))
225	122	uneq2d 3729	. . . . . . . . . . . . . . . . . 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})))
226	223, 224, 225	3eqtr4a 2670	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^nd ‘(1^st ‘𝑇))‘𝑀)}))
227	226	xpeq1d 5062	. . . . . . . . . . . . . . . 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}))
228		xpundir 5095	. . . . . . . . . . . . . . . 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}))
229	227, 228	syl6eq 2660	. . . . . . . . . . . . . . 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})))
230	229	uneq1d 3728	. . . . . . . . . . . . . 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})))
231		un23 3734	. . . . . . . . . . . . . . 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}))
232	231	equncomi 3721	. . . . . . . . . . . . . 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})))
233	230, 232	syl6eq 2660	. . . . . . . . . . . . 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}))))
234	233	fveq1d 6105	. . . . . . . . . . . 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))
235	234	ad2antrr 758	. . . . . . . . . . 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))
236		fnconstg 6006	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ V → ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)})
237	48, 236	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘𝑀)} × {1}) Fn {((2^nd ‘(1^st ‘𝑇))‘𝑀)}
238		fvun2 6180	. . . . . . . . . . . . . . 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))
239	237, 238	mp3an1 1403	. . . . . . . . . . . . . 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))
240	194, 239	mpanr1 715	. . . . . . . . . . . . 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))
241	190, 193, 240	syl2an 493	. . . . . . . . . . . 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))
242	241	anassrs 678	. . . . . . . . . . 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))
243	235, 242	eqtrd 2644	. . . . . . . . . 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))
244	47, 222, 112, 112, 113, 114, 243	ofval 6804	. . . . . . . . 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)))
245	203, 244	eqtr4d 2647	. . . . . . . 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))
246	245	an32s 842	. . . . . . 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))
247	246	anasss 677	. . . . . 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))
248		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
249	248	breq2d 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
250	249	ifbid 4058	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
251	250	csbeq1d 3506	. . . . . . . . . . . . . . 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}))))
252		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (1^st ‘𝑡) = (1^st ‘𝑇))
253	252	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
254	252	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
255	254	imaeq1d 5384	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
256	255	xpeq1d 5062	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
257	254	imaeq1d 5384	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
258	257	xpeq1d 5062	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
259	256, 258	uneq12d 3730	. . . . . . . . . . . . . . . . 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})))
260	253, 259	oveq12d 6567	. . . . . . . . . . . . . . . 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}))))
261	260	csbeq2dv 3944	. . . . . . . . . . . . . . 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}))))
262	251, 261	eqtrd 2644	. . . . . . . . . . . . . 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}))))
263	262	mpteq2dv 4673	. . . . . . . . . . . . 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})))))
264	263	eqeq2d 2620	. . . . . . . . . . . 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}))))))
265	264, 3	elrab2 3333	. . . . . . . . . . 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}))))))
266	265	simprbi 479	. . . . . . . . . 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})))))
267	1, 266	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})))))
268		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < (2^nd ‘𝑇)))
269		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
270	268, 269	ifbieq1d 4059	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑀 − 1) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2^nd ‘𝑇), (𝑀 − 1), (𝑦 + 1)))
271	25	ltm1d 10835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) < (2^nd ‘𝑇))
272	62, 24, 25, 90, 271	lelttrd 10074	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 < (2^nd ‘𝑇))
273	138, 62, 25, 63, 272	lttrd 10077	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 − 1) < (2^nd ‘𝑇))
274	273	iftrued 4044	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑀 − 1) < (2^nd ‘𝑇), (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
275	270, 274	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1))
276	275	csbeq1d 3506	. . . . . . . . . 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 − 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}))))
277		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
278	277	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑀 − 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))))
279	278	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑀 − 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}))
280	279	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}))
281		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
282	281, 74	sylan9eqr 2666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
283	282	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
284	283	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
285	284	xpeq1d 5062	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))
286	280, 285	uneq12d 3730	. . . . . . . . . . . . 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})))
287	286	oveq2d 6565	. . . . . . . . . . . 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}))))
288	79, 287	csbied 3526	. . . . . . . . . . 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}))))
289	288	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋(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}))))
290	276, 289	eqtrd 2644	. . . . . . . . 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))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
291		1red 9934	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
292	62, 26, 291, 91	lesub1dd 10522	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1))
293		elfz2nn0 12300	. . . . . . . . . 10 ⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈ ℕ₀ ∧ (𝑁 − 1) ∈ ℕ₀ ∧ (𝑀 − 1) ≤ (𝑁 − 1)))
294	79, 29, 292, 293	syl3anbrc 1239	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
295		ovex 6577	. . . . . . . . . 10 ⊢ ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V
296	295	a1i 11	. . . . . . . . 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)
297	267, 290, 294, 296	fvmptd 6197	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0}))))
298	297	fveq1d 6105	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(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})))‘𝑛))
299	298	adantr 480	. . . . . 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))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛))
300		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (𝑦 < (2^nd ‘𝑇) ↔ 𝑀 < (2^nd ‘𝑇)))
301		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀)
302	300, 301	ifbieq1d 4059	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^nd ‘𝑇), 𝑀, (𝑦 + 1)))
303	272	iftrued 4044	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑀 < (2^nd ‘𝑇), 𝑀, (𝑦 + 1)) = 𝑀)
304	302, 303	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
305	304	csbeq1d 3506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋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}))))
306		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
307	306	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
308	307	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}))
309		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
310	309	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
311	310	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
312	311	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
313	308, 312	uneq12d 3730	. . . . . . . . . . . . . 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})))
314	313	oveq2d 6565	. . . . . . . . . . . . 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}))))
315	314	adantl 481	. . . . . . . . . . . 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}))))
316	40, 315	csbied 3526	. . . . . . . . . . 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}))))
317	316	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋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}))))
318	305, 317	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋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}))))
319	29	nn0zd 11356	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
320	25, 26, 291, 34	lesub1dd 10522	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘𝑇) − 1) ≤ (𝑁 − 1))
321		eluz2 11569	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) ↔ (((2^nd ‘𝑇) − 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ ((2^nd ‘𝑇) − 1) ≤ (𝑁 − 1)))
322	21, 319, 320, 321	syl3anbrc 1239	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)))
323		fzss2 12252	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘((2^nd ‘𝑇) − 1)) → (1...((2^nd ‘𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
324	322, 323	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1...((2^nd ‘𝑇) − 1)) ⊆ (1...(𝑁 − 1)))
325		1eluzge0 11608	. . . . . . . . . . . 12 ⊢ 1 ∈ (ℤ_≥‘0)
326		fzss1 12251	. . . . . . . . . . . 12 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
327	325, 326	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
328	324, 327	syl6ss 3580	. . . . . . . . . 10 ⊢ (𝜑 → (1...((2^nd ‘𝑇) − 1)) ⊆ (0...(𝑁 − 1)))
329	328, 40	sseldd 3569	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1)))
330		ovex 6577	. . . . . . . . . 10 ⊢ ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V
331	330	a1i 11	. . . . . . . . 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)
332	267, 318, 329, 331	fvmptd 6197	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑀) = ((1^st ‘(1^st ‘𝑇)) ∘_𝑓 + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
333	332	fveq1d 6105	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑀)‘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))
334	333	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → ((𝐹‘𝑀)‘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))
335	247, 299, 334	3eqtr4d 2654	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘𝑀)‘𝑛))
336	335	expr 641	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘𝑀)‘𝑛)))
337	336	necon1d 2804	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) → 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
338		elmapi 7765	. . . . . . . . . . 11 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_𝑚 (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
339	44, 338	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
340	339, 42	ffvelrnd 6268	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ (0..^𝐾))
341		elfzonn0 12380	. . . . . . . . 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)
342	340, 341	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℕ₀)
343	342	nn0red 11229	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℝ)
344	343	ltp1d 10833	. . . . . . 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))
345	343, 344	ltned 10052	. . . . . 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))
346	297	fveq1d 6105	. . . . . . 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))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
347	111	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ V)
348		eqidd 2611	. . . . . . . . 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)))
349		fzss1 12251	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁))
350	76, 349	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁))
351		eluzfz1 12219	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
352	93, 351	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
353		fnfvima 6400	. . . . . . . . . . . . 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)))
354	120, 350, 352, 353	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)))
355		fvun2 6180	. . . . . . . . . . . . 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)))
356	50, 53, 355	mp3an12 1406	. . . . . . . . . . . 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)))
357	69, 354, 356	syl2anc 691	. . . . . . . . . . 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)))
358	51	fvconst2 6374	. . . . . . . . . . . 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)
359	354, 358	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 0)
360	357, 359	eqtrd 2644	. . . . . . . . . 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)
361	360	adantr 480	. . . . . . . . 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)
362	46, 109, 347, 347, 113, 348, 361	ofval 6804	. . . . . . . 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))
363	42, 362	mpdan 699	. . . . . . 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))
364	342	nn0cnd 11230	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ∈ ℂ)
365	364	addid1d 10115	. . . . . . 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)))
366	346, 363, 365	3eqtrd 2648	. . . . . 6 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
367	332	fveq1d 6105	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑀)‘((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)))
368		fzss2 12252	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (1...𝑀) ⊆ (1...𝑁))
369	93, 368	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ⊆ (1...𝑁))
370		elfz1end 12242	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
371	61, 370	sylib 207	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
372		fnfvima 6400	. . . . . . . . . . . . 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)))
373	120, 369, 371, 372	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑀) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
374		fvun1 6179	. . . . . . . . . . . . 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)))
375	205, 134, 374	mp3an12 1406	. . . . . . . . . . . 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)))
376	213, 373, 375	syl2anc 691	. . . . . . . . . . 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)))
377	48	fvconst2 6374	. . . . . . . . . . . 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)
378	373, 377	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = 1)
379	376, 378	eqtrd 2644	. . . . . . . . . 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)
380	379	adantr 480	. . . . . . . . 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)
381	46, 221, 347, 347, 113, 348, 380	ofval 6804	. . . . . . . 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))
382	42, 381	mpdan 699	. . . . . . 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))
383	367, 382	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑀)‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) + 1))
384	345, 366, 383	3netr4d 2859	. . . . 5 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ≠ ((𝐹‘𝑀)‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
385		fveq2 6103	. . . . . 6 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
386		fveq2 6103	. . . . . 6 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘((2^nd ‘(1^st ‘𝑇))‘𝑀)))
387	385, 386	neeq12d 2843	. . . . 5 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘((2^nd ‘(1^st ‘𝑇))‘𝑀)) ≠ ((𝐹‘𝑀)‘((2^nd ‘(1^st ‘𝑇))‘𝑀))))
388	384, 387	syl5ibrcom 236	. . . 4 ⊢ (𝜑 → (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)))
389	388	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)))
390	337, 389	impbid 201	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑀)))
391	42, 390	riota5 6536	1 ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) = ((2^nd ‘(1^st ‘𝑇))‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 {crab 2900 Vcvv 3173 ⦋csb 3499 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ∘_𝑓 cof 6793 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334

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

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

This theorem is referenced by: poimirlem8 32587

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