1smat1 - Mathbox for Thierry Arnoux

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Theorem 1smat1 31422

Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 21434. (Contributed by Thierry Arnoux, 19-Aug-2020.)

**Hypotheses**
Ref	Expression
1smat1.1	⊢ 1 = (1_r‘((1...𝑁) Mat 𝑅))
1smat1.r	⊢ (𝜑 → 𝑅 ∈ Ring)
1smat1.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
1smat1.i	⊢ (𝜑 → 𝐼 ∈ (1...𝑁))

**Assertion**
Ref	Expression
1smat1	⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1_r‘((1...(𝑁 − 1)) Mat 𝑅)))

Proof of Theorem 1smat1 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ (𝐼(subMat1‘ 1 )𝐼) = (𝐼(subMat1‘ 1 )𝐼)
2		1smat1.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
4		1smat1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
5	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
6		1smat1.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		fzfi 13510	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
8		eqid 2736	. . . . . . . . 9 ⊢ ((1...𝑁) Mat 𝑅) = ((1...𝑁) Mat 𝑅)
9		eqid 2736	. . . . . . . . 9 ⊢ (Base‘((1...𝑁) Mat 𝑅)) = (Base‘((1...𝑁) Mat 𝑅))
10		1smat1.1	. . . . . . . . 9 ⊢ 1 = (1_r‘((1...𝑁) Mat 𝑅))
11	8, 9, 10	mat1bas 21300	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1...𝑁) ∈ Fin) → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
12	6, 7, 11	sylancl 589	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
13		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	8, 13	matbas2 21272	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
15	7, 6, 14	sylancr 590	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
16	12, 15	eleqtrrd 2834	. . . . . 6 ⊢ (𝜑 → 1 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
17	16	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 1 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
18		fz1ssnn 13108	. . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
19		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
20	18, 19	sseldi 3885	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
21		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
22	18, 21	sseldi 3885	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
23		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
24		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)))
25	1, 3, 3, 5, 5, 17, 20, 22, 23, 24	smatlem 31415	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) 1 if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
26		eqid 2736	. . . . 5 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
27		eqid 2736	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
28	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...𝑁) ∈ Fin)
29	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑅 ∈ Ring)
30		nnuz 12442	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
31	20, 30	eleqtrdi 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (ℤ_≥‘1))
32		fznatpl1 13131	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → (𝑖 + 1) ∈ (1...𝑁))
33	3, 19, 32	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 + 1) ∈ (1...𝑁))
34		peano2fzr 13090	. . . . . . . 8 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ (𝑖 + 1) ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
35	31, 33, 34	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
36	35, 33	jca 515	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)))
37		eleq1 2818	. . . . . . 7 ⊢ (𝑖 = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → (𝑖 ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
38		eleq1 2818	. . . . . . 7 ⊢ ((𝑖 + 1) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → ((𝑖 + 1) ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
39	37, 38	ifboth 4464	. . . . . 6 ⊢ ((𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁))
40	36, 39	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁))
41	22, 30	eleqtrdi 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (ℤ_≥‘1))
42		fznatpl1 13131	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝑗 + 1) ∈ (1...𝑁))
43	3, 21, 42	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 + 1) ∈ (1...𝑁))
44		peano2fzr 13090	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘1) ∧ (𝑗 + 1) ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
45	41, 43, 44	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
46	45, 43	jca 515	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)))
47		eleq1 2818	. . . . . . 7 ⊢ (𝑗 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → (𝑗 ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
48		eleq1 2818	. . . . . . 7 ⊢ ((𝑗 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → ((𝑗 + 1) ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
49	47, 48	ifboth 4464	. . . . . 6 ⊢ ((𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁))
50	46, 49	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁))
51	8, 26, 27, 28, 29, 40, 50, 10	mat1ov 21299	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) 1 if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))) = if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1_r‘𝑅), (0_g‘𝑅)))
52		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → 𝑖 < 𝐼)
53	52	iftrued 4433	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = 𝑖)
54	53	eqeq1d 2738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
55		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
56	55	iftrued 4433	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
57	56	eqeq2d 2747	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
58		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
59	58	iffalsed 4436	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = (𝑗 + 1))
60	59	eqeq2d 2747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = (𝑗 + 1)))
61	20	nnred 11810	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℝ)
62	61	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
63		fz1ssnn 13108	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ⊆ ℕ
64	63, 4	sseldi 3885	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
65	64	nnred 11810	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
66	65	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
67	22	nnred 11810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℝ)
68	67	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
69		1red 10799	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 1 ∈ ℝ)
70	68, 69	readdcld 10827	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑗 + 1) ∈ ℝ)
71	52	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝐼)
72	64	nnzd 12246	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℤ)
73	72	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
74	22	nnzd 12246	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℤ)
75	74	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℤ)
76	66, 68, 58	nltled 10947	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ≤ 𝑗)
77		zleltp1 12193	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐼 ≤ 𝑗 ↔ 𝐼 < (𝑗 + 1)))
78	77	biimpa 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ 𝐼 ≤ 𝑗) → 𝐼 < (𝑗 + 1))
79	73, 75, 76, 78	syl21anc 838	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 < (𝑗 + 1))
80	62, 66, 70, 71, 79	lttrd 10958	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < (𝑗 + 1))
81	62, 80	ltned 10933	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ (𝑗 + 1))
82	81	neneqd 2937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = (𝑗 + 1))
83	62, 66, 68, 71, 76	ltletrd 10957	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝑗)
84	62, 83	ltned 10933	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
85	84	neneqd 2937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
86	82, 85	2falsed 380	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = (𝑗 + 1) ↔ 𝑖 = 𝑗))
87	60, 86	bitrd 282	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
88	57, 87	pm2.61dan 813	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
89	54, 88	bitrd 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
90		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ¬ 𝑖 < 𝐼)
91	90	iffalsed 4436	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = (𝑖 + 1))
92	91	eqeq1d 2738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
93		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
94	93	iftrued 4433	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
95	94	eqeq2d 2747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = 𝑗))
96	67	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
97	65	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
98	61	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
99		1red 10799	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 1 ∈ ℝ)
100	98, 99	readdcld 10827	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ∈ ℝ)
101	72	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
102	20	nnzd 12246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℤ)
103	102	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℤ)
104	90	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 < 𝐼)
105	97, 98, 104	nltled 10947	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ≤ 𝑖)
106		zleltp1 12193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝐼 ≤ 𝑖 ↔ 𝐼 < (𝑖 + 1)))
107	106	biimpa 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝐼 ≤ 𝑖) → 𝐼 < (𝑖 + 1))
108	101, 103, 105, 107	syl21anc 838	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 < (𝑖 + 1))
109	96, 97, 100, 93, 108	lttrd 10958	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < (𝑖 + 1))
110	96, 109	ltned 10933	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ (𝑖 + 1))
111	110	necomd 2987	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ≠ 𝑗)
112	111	neneqd 2937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ (𝑖 + 1) = 𝑗)
113	96, 97, 98, 93, 105	ltletrd 10957	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝑖)
114	96, 113	ltned 10933	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ 𝑖)
115	114	necomd 2987	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
116	115	neneqd 2937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
117	112, 116	2falsed 380	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = 𝑗 ↔ 𝑖 = 𝑗))
118	95, 117	bitrd 282	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
119		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
120	119	iffalsed 4436	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = (𝑗 + 1))
121	120	eqeq2d 2747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = (𝑗 + 1)))
122	20	nncnd 11811	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℂ)
123	122	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 ∈ ℂ)
124	22	nncnd 11811	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℂ)
125	124	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑗 ∈ ℂ)
126		1cnd 10793	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 1 ∈ ℂ)
127		simpr 488	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → (𝑖 + 1) = (𝑗 + 1))
128	123, 125, 126, 127	addcan2ad 11003	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 = 𝑗)
129		simpr 488	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
130	129	oveq1d 7206	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → (𝑖 + 1) = (𝑗 + 1))
131	128, 130	impbida 801	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = (𝑗 + 1) ↔ 𝑖 = 𝑗))
132	121, 131	bitrd 282	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
133	118, 132	pm2.61dan 813	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
134	92, 133	bitrd 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
135	89, 134	pm2.61dan 813	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
136	135	ifbid 4448	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1_r‘𝑅), (0_g‘𝑅)) = if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅)))
137		eqid 2736	. . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
138		fzfid 13511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ∈ Fin)
139		eqid 2736	. . . . . 6 ⊢ (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) = (1_r‘((1...(𝑁 − 1)) Mat 𝑅))
140	137, 26, 27, 138, 29, 19, 21, 139	mat1ov 21299	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅)))
141	136, 140	eqtr4d 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1_r‘𝑅), (0_g‘𝑅)) = (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
142	25, 51, 141	3eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
143	142	ralrimivva 3102	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
144	1, 2, 2, 4, 4, 16	smatrcl 31414	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
145		elmapfn 8524	. . . 4 ⊢ ((𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → (𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
146	144, 145	syl 17	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
147		fzfi 13510	. . . . . 6 ⊢ (1...(𝑁 − 1)) ∈ Fin
148		eqid 2736	. . . . . . 7 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	137, 148, 139	mat1bas 21300	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1...(𝑁 − 1)) ∈ Fin) → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
150	6, 147, 149	sylancl 589	. . . . 5 ⊢ (𝜑 → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
151	137, 13	matbas2 21272	. . . . . 6 ⊢ (((1...(𝑁 − 1)) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
152	147, 6, 151	sylancr 590	. . . . 5 ⊢ (𝜑 → ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
153	150, 152	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
154		elmapfn 8524	. . . 4 ⊢ ((1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
155	153, 154	syl 17	. . 3 ⊢ (𝜑 → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
156		eqfnov2 7318	. . 3 ⊢ (((𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))) ∧ (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → ((𝐼(subMat1‘ 1 )𝐼) = (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
157	146, 155, 156	syl2anc 587	. 2 ⊢ (𝜑 → ((𝐼(subMat1‘ 1 )𝐼) = (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1_r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
158	143, 157	mpbird 260	1 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1_r‘((1...(𝑁 − 1)) Mat 𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ifcif 4425 class class class wbr 5039 × cxp 5534 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ↑_m cmap 8486 Fincfn 8604 ℂcc 10692 ℝcr 10693 1c1 10695 + caddc 10697 < clt 10832 ≤ cle 10833 − cmin 11027 ℕcn 11795 ℤcz 12141 ℤ_≥cuz 12403 ...cfz 13060 Basecbs 16666 0_gc0g 16898 1_rcur 19470 Ringcrg 19516 Mat cmat 21258 subMat1csmat 31411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-sra 20163 df-rgmod 20164 df-dsmm 20648 df-frlm 20663 df-mamu 21237 df-mat 21259 df-smat 31412

This theorem is referenced by: (None)

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