madjusmdetlem2 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem2 31181

Description: Lemma for madjusmdet 31184. (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 12269	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2900	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		eluzfz2 12910	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
6		eqid 2798	. . . . . . . . . . 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 2798	. . . . . . . . . . 11 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
9		eqid 2798	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
10	6, 7, 8, 9	fzto1st 30795	. . . . . . . . . 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 18495	. . . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
15		fznatpl1 12956	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
16	1, 15	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ∈ (1...𝑁))
17		eqeq1 2802	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → (𝑖 = 1 ↔ 𝑥 = 1))
18		breq1 5033	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
19		oveq1 7142	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → (𝑖 − 1) = (𝑥 − 1))
20		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → 𝑖 = 𝑥)
21	18, 19, 20	ifbieq12d 4452	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))
22	17, 21	ifbieq2d 4450	. . . . . . . . . . . 12 ⊢ (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 5133	. . . . . . . . . . 11 ⊢ (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 2821	. . . . . . . . . 10 ⊢ 𝑆 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)))
25	24	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑆 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))))
26		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
27		1red 10631	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 ∈ ℝ)
28		fz1ssnn 12933	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
29		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ (1...(𝑁 − 1)))
30	28, 29	sseldi 3913	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ)
31	30	nnrpd 12417	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℝ⁺)
32	31	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℝ⁺)
33	27, 32	ltaddrp2d 12453	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < (𝑋 + 1))
34	27, 33	ltned 10765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 ≠ (𝑋 + 1))
35	34	necomd 3042	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≠ 1)
36	26, 35	eqnetrd 3054	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
37	36	neneqd 2992	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
38	37	iffalsed 4436	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)) = if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥))
39	1	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
40	30	nnnn0d 11943	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℕ₀)
41	39	nnnn0d 11943	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
42		elfzle2 12906	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (1...(𝑁 − 1)) → 𝑋 ≤ (𝑁 − 1))
43	29, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ≤ (𝑁 − 1))
44		nn0ltlem1 12030	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑋 < 𝑁 ↔ 𝑋 ≤ (𝑁 − 1)))
45	44	biimpar 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑋 ≤ (𝑁 − 1)) → 𝑋 < 𝑁)
46	40, 41, 43, 45	syl21anc 836	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 < 𝑁)
47		nnltp1le 12026	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑋 < 𝑁 ↔ (𝑋 + 1) ≤ 𝑁))
48	47	biimpa 480	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑋 < 𝑁) → (𝑋 + 1) ≤ 𝑁)
49	30, 39, 46, 48	syl21anc 836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑋 + 1) ≤ 𝑁)
50	49	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝑁)
51	26, 50	eqbrtrd 5052	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝑁)
52	51	iftrued 4433	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥) = (𝑥 − 1))
53	26	oveq1d 7150	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = ((𝑋 + 1) − 1))
54	30	nncnd 11641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ)
55		1cnd 10625	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 1 ∈ ℂ)
56	54, 55	pncand 10987	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑋 + 1) − 1) = 𝑋)
57	56	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ((𝑋 + 1) − 1) = 𝑋)
58	53, 57	eqtrd 2833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
59	38, 52, 58	3eqtrd 2837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝑁, if(𝑥 ≤ 𝑁, (𝑥 − 1), 𝑥)) = 𝑋)
60	25, 59, 16, 29	fvmptd 6752	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑆‘(𝑋 + 1)) = 𝑋)
61	60	idi 1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑆‘(𝑋 + 1)) = 𝑋)
62		f1ocnvfv 7013	. . . . . . . 8 ⊢ ((𝑆:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) → ((𝑆‘(𝑋 + 1)) = 𝑋 → (^◡𝑆‘𝑋) = (𝑋 + 1)))
63	62	imp 410	. . . . . . 7 ⊢ (((𝑆:(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑋 + 1) ∈ (1...𝑁)) ∧ (𝑆‘(𝑋 + 1)) = 𝑋) → (^◡𝑆‘𝑋) = (𝑋 + 1))
64	14, 16, 61, 63	syl21anc 836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (^◡𝑆‘𝑋) = (𝑋 + 1))
65	64	fveq2d 6649	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
66	65	adantr 484	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
67		madjusmdetlem2.p	. . . . . . 7 ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
68	20	breq1d 5040	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (𝑖 ≤ 𝐼 ↔ 𝑥 ≤ 𝐼))
69	68, 19, 20	ifbieq12d 4452	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖) = if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))
70	17, 69	ifbieq2d 4450	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
71	70	cbvmptv 5133	. . . . . . 7 ⊢ (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)))
72	67, 71	eqtri 2821	. . . . . 6 ⊢ 𝑃 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)))
73	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑃 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))))
74	33, 26	breqtrrd 5058	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 < 𝑥)
75	27, 74	ltned 10765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 1 ≠ 𝑥)
76	75	necomd 3042	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≠ 1)
77	76	neneqd 2992	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 = 1)
78	77	iffalsed 4436	. . . . . . 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))
79	78	adantlr 714	. . . . . 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))
80		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
81	30	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 ∈ ℕ)
82		fz1ssnn 12933	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
83		madjusmdet.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
84	82, 83	sseldi 3913	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
85	84	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝐼 ∈ ℕ)
86		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑋 < 𝐼)
87		nnltp1le 12026	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝑋 < 𝐼 ↔ (𝑋 + 1) ≤ 𝐼))
88	87	biimpa 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ≤ 𝐼)
89	81, 85, 86, 88	syl21anc 836	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑋 + 1) ≤ 𝐼)
90	80, 89	eqbrtrd 5052	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 ≤ 𝐼)
91	90	iftrued 4433	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = (𝑥 − 1))
92	58	adantlr 714	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 − 1) = 𝑋)
93	79, 91, 92	3eqtrd 2837	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = 𝑋)
94	16	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
95		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...(𝑁 − 1)))
96	73, 93, 94, 95	fvmptd 6752	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = 𝑋)
97	66, 96	eqtr2d 2834	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑋 < 𝐼) → 𝑋 = (𝑃‘(^◡𝑆‘𝑋)))
98	65	adantr 484	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(^◡𝑆‘𝑋)) = (𝑃‘(𝑋 + 1)))
99	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → 𝑃 = (𝑥 ∈ (1...𝑁) ↦ if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥))))
100	78	adantlr 714	. . . . . 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))
101	30	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 ∈ ℕ)
102	84	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝐼 ∈ ℕ)
103	26	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 = (𝑋 + 1))
104		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑥 ≤ 𝐼)
105	103, 104	eqbrtrrd 5054	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → (𝑋 + 1) ≤ 𝐼)
106	87	biimpar 481	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ (𝑋 + 1) ≤ 𝐼) → 𝑋 < 𝐼)
107	101, 102, 105, 106	syl21anc 836	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ 𝑥 ≤ 𝐼) → 𝑋 < 𝐼)
108	107	ex 416	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (𝑥 ≤ 𝐼 → 𝑋 < 𝐼))
109	108	con3d 155	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) → (¬ 𝑋 < 𝐼 → ¬ 𝑥 ≤ 𝐼))
110	109	imp 410	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ 𝑥 = (𝑋 + 1)) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑥 ≤ 𝐼)
111	110	an32s 651	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → ¬ 𝑥 ≤ 𝐼)
112	111	iffalsed 4436	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = 𝑥)
113		simpr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → 𝑥 = (𝑋 + 1))
114	112, 113	eqtrd 2833	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥) = (𝑋 + 1))
115	100, 114	eqtrd 2833	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝑋 + 1)) → if(𝑥 = 1, 𝐼, if(𝑥 ≤ 𝐼, (𝑥 − 1), 𝑥)) = (𝑋 + 1))
116	16	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ (1...𝑁))
117	99, 115, 116, 116	fvmptd 6752	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑃‘(𝑋 + 1)) = (𝑋 + 1))
118	98, 117	eqtr2d 2834	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) = (𝑃‘(^◡𝑆‘𝑋)))
119	97, 118	ifeqda 4460	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑃‘(^◡𝑆‘𝑋)))
120		f1ocnv 6602	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
121	11, 12, 120	3syl 18	. . . . 5 ⊢ (𝜑 → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
122		f1ofun 6592	. . . . 5 ⊢ (^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑆)
123	121, 122	syl 17	. . . 4 ⊢ (𝜑 → Fun ^◡𝑆)
124	123	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → Fun ^◡𝑆)
125		fzdif2 30540	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
126	3, 125	syl 17	. . . . . . 7 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
127		difss 4059	. . . . . . 7 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
128	126, 127	eqsstrrdi 3970	. . . . . 6 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
129		f1odm 6594	. . . . . . 7 ⊢ (^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → dom ^◡𝑆 = (1...𝑁))
130	121, 129	syl 17	. . . . . 6 ⊢ (𝜑 → dom ^◡𝑆 = (1...𝑁))
131	128, 130	sseqtrrd 3956	. . . . 5 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ dom ^◡𝑆)
132	131	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → (1...(𝑁 − 1)) ⊆ dom ^◡𝑆)
133	132, 29	sseldd 3916	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ dom ^◡𝑆)
134		fvco 6736	. . 3 ⊢ ((Fun ^◡𝑆 ∧ 𝑋 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
135	124, 133, 134	syl2anc 587	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → ((𝑃 ∘ ^◡𝑆)‘𝑋) = (𝑃‘(^◡𝑆‘𝑋)))
136	119, 135	eqtr4d 2836	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 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ifcif 4425 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ^◡ccnv 5518 dom cdm 5519 ∘ ccom 5523 Fun wfun 6318 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℕ₀cn0 11885 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 Basecbs 16475 ._rcmulr 16558 SymGrpcsymg 18487 CRingccrg 19291 ℤRHomczrh 20193 Mat cmat 21012 maDet cmdat 21189 maAdju cmadu 21237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-efmnd 18026 df-symg 18488 df-pmtr 18562

This theorem is referenced by: madjusmdetlem3 31182

W3C validator