madjusmdetlem2 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem2 33818

Description: Lemma for madjusmdet 33821. (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 12836	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		eluzfz2 13493	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
6		eqid 2729	. . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
9		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
10	6, 7, 8, 9	fzto1st 33060	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
11	5, 10	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
12	8, 9	symgbasf1o 19305	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
15		fznatpl1 13539	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
16	1, 15	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
17		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → (𝑖 = 1 ↔ 𝑥 = 1))
18		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
19		oveq1 7394	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → (𝑖 − 1) = (𝑥 − 1))
20		id 22	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → 𝑖 = 𝑥)
21	18, 19, 20	ifbieq12d 4517	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))
22	17, 21	ifbieq2d 4515	. . . . . . . . . 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 5211	. . . . . . . . 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 2752	. . . . . . . 8 ⊢ 𝑆 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)))
25		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
26		1red 11175	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 ∈ ℝ)
27		fz1ssnn 13516	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
28		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ (1...(𝑁 − 1)))
29	27, 28	sselid 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ)
30	29	nnrpd 12993	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℝ⁺)
31	30	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℝ⁺)
32	26, 31	ltaddrp2d 13029	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < (𝑋 + 1))
33	26, 32	gtned 11309	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≠ 1)
34	25, 33	eqnetrd 2992	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
35	34	neneqd 2930	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
36	35	iffalsed 4499	. . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
38	29	nnnn0d 12503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ₀)
39	37	nnnn0d 12503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
40		elfzle2 13489	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (1...(𝑁 − 1)) → 𝑋 ≤ (𝑁 − 1))
41	28, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ≤ (𝑁 − 1))
42		nn0ltlem1 12594	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑋 < 𝑁 ↔ 𝑋 ≤ (𝑁 − 1)))
43	42	biimpar 477	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑋 ≤ (𝑁 − 1)) → 𝑋 < 𝑁)
44	38, 39, 41, 43	syl21anc 837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 < 𝑁)
45		nnltp1le 12590	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑋 < 𝑁 ↔ (𝑋 + 1) ≤ 𝑁))
46	45	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑋 < 𝑁) → (𝑋 + 1) ≤ 𝑁)
47	29, 37, 44, 46	syl21anc 837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ≤ 𝑁)
48	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝑁)
49	25, 48	eqbrtrd 5129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝑁)
50	49	iftrued 4496	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥) = (𝑥 − 1))
51	25	oveq1d 7402	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = ((𝑋 + 1) − 1))
52	29	nncnd 12202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ)
53		1cnd 11169	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 1 ∈ ℂ)
54	52, 53	pncand 11534	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑋 + 1) − 1) = 𝑋)
55	54	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ((𝑋 + 1) − 1) = 𝑋)
56	51, 55	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
57	36, 50, 56	3eqtrd 2768	. . . . . . . 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 6976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑆‘(𝑋 + 1)) = 𝑋)
59		f1ocnvfv 7253	. . . . . . . 8 ⊢ ((𝑆:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) → ((𝑆‘(𝑋 + 1)) = 𝑋 → (^◡𝑆‘𝑋) = (𝑋 + 1)))
60	59	imp 406	. . . . . . 7 ⊢ (((𝑆:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) ∧ (𝑆‘(𝑋 + 1)) = 𝑋) → (^◡𝑆‘𝑋) = (𝑋 + 1))
61	14, 16, 58, 60	syl21anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (^◡𝑆‘𝑋) = (𝑋 + 1))
62	61	fveq2d 6862	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
63	62	adantr 480	. . . 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 5110	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝐼 ↔ 𝑥 ≤ 𝐼))
66	65, 19, 20	ifbieq12d 4517	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
67	17, 66	ifbieq2d 4515	. . . . . . 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 5211	. . . . . 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 2752	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
70	32, 25	breqtrrd 5135	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < 𝑥)
71	26, 70	gtned 11309	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
72	71	neneqd 2930	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
73	72	iffalsed 4499	. . . . . . 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 715	. . . . . 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 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
76	29	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℕ)
77		fz1ssnn 13516	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
78		madjusmdet.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
79	77, 78	sselid 3944	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
80	79	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝐼 ∈ ℕ)
81		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 < 𝐼)
82		nnltp1le 12590	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝑋 < 𝐼 ↔ (𝑋 + 1) ≤ 𝐼))
83	82	biimpa 476	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ≤ 𝐼)
84	76, 80, 81, 83	syl21anc 837	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝐼)
85	75, 84	eqbrtrd 5129	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝐼)
86	85	iftrued 4496	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = (𝑥 − 1))
87	56	adantlr 715	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
88	74, 86, 87	3eqtrd 2768	. . . . 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 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
90		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...(𝑁 − 1)))
91	69, 88, 89, 90	fvmptd2 6976	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = 𝑋)
92	63, 91	eqtr2d 2765	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 = (𝑃‘(^◡𝑆‘𝑋)))
93	62	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
94	73	adantlr 715	. . . . . 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 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 ∈ ℕ)
96	79	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝐼 ∈ ℕ)
97		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 = (𝑋 + 1))
98		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 ≤ 𝐼)
99	97, 98	eqbrtrrd 5131	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → (𝑋 + 1) ≤ 𝐼)
100	82	biimpar 477	. . . . . . . . . 10 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ (𝑋 + 1) ≤ 𝐼) → 𝑋 < 𝐼)
101	95, 96, 99, 100	syl21anc 837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 < 𝐼)
102	101	stoic1a 1772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑥 ≤ 𝐼)
103	102	an32s 652	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 ≤ 𝐼)
104	103	iffalsed 4499	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = 𝑥)
105		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
106	94, 104, 105	3eqtrd 2768	. . . . 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 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
108	69, 106, 107, 107	fvmptd2 6976	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = (𝑋 + 1))
109	93, 108	eqtr2d 2765	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) = (𝑃‘(^◡𝑆‘𝑋)))
110	92, 109	ifeqda 4525	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑃‘(^◡𝑆‘𝑋)))
111		f1ocnv 6812	. . . . 5 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
112	11, 12, 111	3syl 18	. . . 4 ⊢ (𝜑 → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
113		f1ofun 6802	. . . 4 ⊢ (^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑆)
114	112, 113	syl 17	. . 3 ⊢ (𝜑 → Fun ^◡𝑆)
115		fzdif2 32713	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
116	3, 115	syl 17	. . . . . 6 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
117		difss 4099	. . . . . 6 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
118	116, 117	eqsstrrdi 3992	. . . . 5 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
119		f1odm 6804	. . . . . 6 ⊢ (^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → dom ^◡𝑆 = (1...𝑁))
120	112, 119	syl 17	. . . . 5 ⊢ (𝜑 → dom ^◡𝑆 = (1...𝑁))
121	118, 120	sseqtrrd 3984	. . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ dom ^◡𝑆)
122	121	sselda 3946	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ dom ^◡𝑆)
123		fvco 6959	. . 3 ⊢ ((Fun ^◡𝑆 ∧ 𝑋 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
124	114, 122, 123	syl2an2r 685	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
125	110, 124	eqtr4d 2767	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ifcif 4488 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 ^◡ccnv 5637 dom cdm 5638 ∘ ccom 5642 Fun wfun 6505 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 ℕ₀cn0 12442 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 Basecbs 17179 ._rcmulr 17221 SymGrpcsymg 19299 CRingccrg 20143 ℤRHomczrh 21409 Mat cmat 22294 maDet cmdat 22471 maAdju cmadu 22519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-tset 17239 df-efmnd 18796 df-symg 19300 df-pmtr 19372

This theorem is referenced by: madjusmdetlem3 33819

W3C validator