madjusmdetlem2 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem2 34527

Description: Lemma for madjusmdet 34530. (Contributed by Thierry Arnoux, 26-Aug-2020.)

**Hypotheses**
Ref	Expression
madjusmdet.b	⊢ 𝐵 = (Base‘𝐴)
madjusmdet.a	⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
madjusmdet.d	⊢ 𝐷 = ((1...𝑁) maDet 𝑅)
madjusmdet.k	⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)
madjusmdet.t	⊢ · = (._r‘𝑅)
madjusmdet.z	⊢ 𝑍 = (ℤRHom‘𝑅)
madjusmdet.e	⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
madjusmdet.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
madjusmdet.r	⊢ (𝜑 → 𝑅 ∈ CRing)
madjusmdet.i	⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
madjusmdet.j	⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
madjusmdet.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
madjusmdetlem2.p	⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
madjusmdetlem2.s	⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))

**Assertion**
Ref	Expression
madjusmdetlem2	⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑋))

Distinct variable groups: 𝐵,𝑖 𝑖,𝐼 𝑖,𝐽 𝑖,𝑀 𝑖,𝑁 𝑃,𝑖 𝑅,𝑖 𝜑,𝑖 𝑆,𝑖 Allowed substitution hints: 𝐴(𝑖) 𝐷(𝑖) · (𝑖) 𝐸(𝑖) 𝐾(𝑖) 𝑋(𝑖) 𝑍(𝑖)

Proof of Theorem madjusmdetlem2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madjusmdet.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		nnuz 13028	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		eluzfz2 13689	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
5	3, 4	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
6		eqid 2763	. . . . . . . . . . 11 ⊢ (1...𝑁) = (1...𝑁)
7		madjusmdetlem2.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))
8		eqid 2763	. . . . . . . . . . 11 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
9		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
10	6, 7, 8, 9	fzto1st 33731	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
11	5, 10	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
12	8, 9	symgbasf1o 19616	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
13	11, 12	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
14	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
15		fznatpl1 13736	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
16	1, 15	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
17		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → (𝑖 = 1 ↔ 𝑥 = 1))
18		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
19		oveq1 7447	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → (𝑖 − 1) = (𝑥 − 1))
20		id 23	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → 𝑖 = 𝑥)
21	18, 19, 20	ifbieq12d 4514	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))
22	17, 21	ifbieq2d 4512	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) = if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)))
23	22	cbvmptv 5212	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)))
24	7, 23	eqtri 2786	. . . . . . . 8 ⊢ 𝑆 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)))
25		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
26		1red 11333	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 ∈ ℝ)
27		fz1ssnn 13713	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
28		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ (1...(𝑁 − 1)))
29	27, 28	sselid 3911	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ)
30	29	nnrpd 13186	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℝ⁺)
31	30	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℝ⁺)
32	26, 31	ltaddrp2d 13222	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < (𝑋 + 1))
33	26, 32	gtned 11469	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≠ 1)
34	25, 33	eqnetrd 3026	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
35	34	neneqd 2964	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
36	35	iffalsed 4496	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)) = if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))
37	1	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
38	29	nnnn0d 12691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ₀)
39	37	nnnn0d 12691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
40		elfzle2 13685	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (1...(𝑁 − 1)) → 𝑋 ≤ (𝑁 − 1))
41	28, 40	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ≤ (𝑁 − 1))
42		nn0ltlem1 12783	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑋 < 𝑁 ↔ 𝑋 ≤ (𝑁 − 1)))
43	42	biimpar 483	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑋 ≤ (𝑁 − 1)) → 𝑋 < 𝑁)
44	38, 39, 41, 43	syl21anc 851	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 < 𝑁)
45		nnltp1le 12779	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑋 < 𝑁 ↔ (𝑋 + 1) ≤ 𝑁))
46	45	biimpa 482	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑋 < 𝑁) → (𝑋 + 1) ≤ 𝑁)
47	29, 37, 44, 46	syl21anc 851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ≤ 𝑁)
48	47	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝑁)
49	25, 48	eqbrtrd 5130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝑁)
50	49	iftrued 4493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥) = (𝑥 − 1))
51	25	oveq1d 7455	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = ((𝑋 + 1) − 1))
52	29	nncnd 12375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ)
53		1cnd 11326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 1 ∈ ℂ)
54	52, 53	pncand 11694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑋 + 1) − 1) = 𝑋)
55	54	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ((𝑋 + 1) − 1) = 𝑋)
56	51, 55	eqtrd 2798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
57	36, 50, 56	3eqtrd 2802	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)) = 𝑋)
58	24, 57, 16, 28	fvmptd2 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑆‘(𝑋 + 1)) = 𝑋)
59		f1ocnvfv 7306	. . . . . . . 8 ⊢ ((𝑆:(1...𝑁)⟶1-1onto(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) → ((𝑆‘(𝑋 + 1)) = 𝑋 → (^◡𝑆‘𝑋) = (𝑋 + 1)))
60	59	imp 412	. . . . . . 7 ⊢ (((𝑆:(1...𝑁)⟶1-1onto(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) ∧ (𝑆‘(𝑋 + 1)) = 𝑋) → (^◡𝑆‘𝑋) = (𝑋 + 1))
61	14, 16, 58, 60	syl21anc 851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (^◡𝑆‘𝑋) = (𝑋 + 1))
62	61	fveq2d 6909	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
63	62	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
64		madjusmdetlem2.p	. . . . . 6 ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
65		breq1 5109	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝐼 ↔ 𝑥 ≤ 𝐼))
66	65, 19, 20	ifbieq12d 4514	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
67	17, 66	ifbieq2d 4512	. . . . . . 7 ⊢ (𝑖 = 𝑥 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
68	67	cbvmptv 5212	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
69	64, 68	eqtri 2786	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
70	32, 25	breqtrrd 5136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < 𝑥)
71	26, 70	gtned 11469	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
72	71	neneqd 2964	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
73	72	iffalsed 4496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
74	73	adantlr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
75		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
76	29	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℕ)
77		fz1ssnn 13713	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
78		madjusmdet.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
79	77, 78	sselid 3911	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
80	79	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝐼 ∈ ℕ)
81		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 < 𝐼)
82		nnltp1le 12779	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝑋 < 𝐼 ↔ (𝑋 + 1) ≤ 𝐼))
83	82	biimpa 482	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ≤ 𝐼)
84	76, 80, 81, 83	syl21anc 851	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝐼)
85	75, 84	eqbrtrd 5130	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝐼)
86	85	iftrued 4493	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = (𝑥 − 1))
87	56	adantlr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
88	74, 86, 87	3eqtrd 2802	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = 𝑋)
89	16	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
90		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...(𝑁 − 1)))
91	69, 88, 89, 90	fvmptd2 7022	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = 𝑋)
92	63, 91	eqtr2d 2799	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 = (𝑃‘(^◡𝑆‘𝑋)))
93	62	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
94	73	adantlr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
95	29	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 ∈ ℕ)
96	79	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝐼 ∈ ℕ)
97		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 = (𝑋 + 1))
98		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 ≤ 𝐼)
99	97, 98	eqbrtrrd 5132	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → (𝑋 + 1) ≤ 𝐼)
100	82	biimpar 483	. . . . . . . . . 10 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ (𝑋 + 1) ≤ 𝐼) → 𝑋 < 𝐼)
101	95, 96, 99, 100	syl21anc 851	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 < 𝐼)
102	101	stoic1a 1805	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑥 ≤ 𝐼)
103	102	an32s 665	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 ≤ 𝐼)
104	103	iffalsed 4496	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = 𝑥)
105		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
106	94, 104, 105	3eqtrd 2802	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = (𝑋 + 1))
107	16	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
108	69, 106, 107, 107	fvmptd2 7022	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = (𝑋 + 1))
109	93, 108	eqtr2d 2799	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) = (𝑃‘(^◡𝑆‘𝑋)))
110	92, 109	ifeqda 4522	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑃‘(^◡𝑆‘𝑋)))
111		f1ocnv 6856	. . . . 5 ⊢ (𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → ^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
112	11, 12, 111	3syl 19	. . . 4 ⊢ (𝜑 → ^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
113		f1ofun 6845	. . . 4 ⊢ (^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → Fun ^◡𝑆)
114	112, 113	syl 18	. . 3 ⊢ (𝜑 → Fun ^◡𝑆)
115		fzdif2 33449	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
116	3, 115	syl 18	. . . . . 6 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
117		difss 4075	. . . . . 6 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
118	116, 117	eqsstrrdi 3958	. . . . 5 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
119		f1odm 6847	. . . . . 6 ⊢ (^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → dom ^◡𝑆 = (1...𝑁))
120	112, 119	syl 18	. . . . 5 ⊢ (𝜑 → dom ^◡𝑆 = (1...𝑁))
121	118, 120	sseqtrrd 3950	. . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ dom ^◡𝑆)
122	121	sselda 3913	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ dom ^◡𝑆)
123		fvco 7003	. . 3 ⊢ ((Fun ^◡𝑆 ∧ 𝑋 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
124	114, 122, 123	syl2an2r 698	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
125	110, 124	eqtr4d 2801	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∖ cdif 4048 ifcif 4485 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 ^◡ccnv 5654 dom cdm 5655 ∘ ccom 5659 Fun wfun 6549 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 1c1 11225 + caddc 11227 < clt 11367 ≤ cle 11368 − cmin 11565 ℕcn 12359 ℕ₀cn0 12630 ℤ_≥cuz 12989 ℝ⁺crp 13144 ...cfz 13663 Basecbs 17413 ._rcmulr 17455 SymGrpcsymg 19610 CRingccrg 20487 ℤRHomczrh 21832 Mat cmat 22749 maDet cmdat 22926 maAdju cmadu 22974

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-uz 12990 df-rp 13145 df-fz 13664 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-tset 17473 df-efmnd 19091 df-symg 19611 df-pmtr 19683

This theorem is used by: madjusmdetlem3 34528

W3C validator