Theorem 1smat1 33804

Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22590. (Contributed by Thierry Arnoux, 19-Aug-2020.)

Hypotheses

Ref	Expression
1smat1.1	⊢ 1 = (1r‘((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) = (1r‘((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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
4		1smat1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
6		1smat1.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		fzfi 14014	. . . . . . . 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 = (1r‘((1...𝑁) Mat 𝑅))
11	8, 9, 10	mat1bas 22456	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1...𝑁) ∈ Fin) → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
12	6, 7, 11	sylancl 586	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
13		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	8, 13	matbas2 22428	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
15	7, 6, 14	sylancr 587	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
16	12, 15	eleqtrrd 2843	. . . . . 6 ⊢ (𝜑 → 1 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 1 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
18		fz1ssnn 13596	. . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
19		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
20	18, 19	sselid 3980	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
21		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
22	18, 21	sselid 3980	. . . . 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 33797	. . . 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 ⊢ (1r‘𝑅) = (1r‘𝑅)
27		eqid 2736	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...𝑁) ∈ Fin)
29	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑅 ∈ Ring)
30		nnuz 12922	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
31	20, 30	eleqtrdi 2850	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (ℤ≥‘1))
32		fznatpl1 13619	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → (𝑖 + 1) ∈ (1...𝑁))
33	3, 19, 32	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 + 1) ∈ (1...𝑁))
34		peano2fzr 13578	. . . . . . . 8 ⊢ ((𝑖 ∈ (ℤ≥‘1) ∧ (𝑖 + 1) ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
35	31, 33, 34	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
36	35, 33	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)))
37		eleq1 2828	. . . . . . 7 ⊢ (𝑖 = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → (𝑖 ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
38		eleq1 2828	. . . . . . 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 4564	. . . . . 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 2850	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (ℤ≥‘1))
42		fznatpl1 13619	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝑗 + 1) ∈ (1...𝑁))
43	3, 21, 42	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 + 1) ∈ (1...𝑁))
44		peano2fzr 13578	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ≥‘1) ∧ (𝑗 + 1) ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
45	41, 43, 44	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
46	45, 43	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)))
47		eleq1 2828	. . . . . . 7 ⊢ (𝑗 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → (𝑗 ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
48		eleq1 2828	. . . . . . 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 4564	. . . . . 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 22455	. . . 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)), (1r‘𝑅), (0g‘𝑅)))
52		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → 𝑖 < 𝐼)
53	52	iftrued 4532	. . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
56	55	iftrued 4532	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
59	58	iffalsed 4535	. . . . . . . . . . 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 12282	. . . . . . . . . . . . . 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 13596	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ⊆ ℕ
64	63, 4	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
65	64	nnred 12282	. . . . . . . . . . . . . . 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 12282	. . . . . . . . . . . . . . . 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 11263	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 1 ∈ ℝ)
70	68, 69	readdcld 11291	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑗 + 1) ∈ ℝ)
71	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝐼)
72	64	nnzd 12642	. . . . . . . . . . . . . . . 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 12642	. . . . . . . . . . . . . . . 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 11412	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ≤ 𝑗)
77		zleltp1 12670	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐼 ≤ 𝑗 ↔ 𝐼 < (𝑗 + 1)))
78	77	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ 𝐼 ≤ 𝑗) → 𝐼 < (𝑗 + 1))
79	73, 75, 76, 78	syl21anc 837	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 < (𝑗 + 1))
80	62, 66, 70, 71, 79	lttrd 11423	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < (𝑗 + 1))
81	62, 80	ltned 11398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ (𝑗 + 1))
82	81	neneqd 2944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = (𝑗 + 1))
83	62, 66, 68, 71, 76	ltletrd 11422	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝑗)
84	62, 83	ltned 11398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
85	84	neneqd 2944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
86	82, 85	2falsed 376	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = (𝑗 + 1) ↔ 𝑖 = 𝑗))
87	60, 86	bitrd 279	. . . . . . . . 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 812	. . . . . . . 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 279	. . . . . . 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 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ¬ 𝑖 < 𝐼)
91	90	iffalsed 4535	. . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
94	93	iftrued 4532	. . . . . . . . . . 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 11263	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 1 ∈ ℝ)
100	98, 99	readdcld 11291	. . . . . . . . . . . . . . 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 12642	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 < 𝐼)
105	97, 98, 104	nltled 11412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ≤ 𝑖)
106		zleltp1 12670	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝐼 ≤ 𝑖 ↔ 𝐼 < (𝑖 + 1)))
107	106	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝐼 ≤ 𝑖) → 𝐼 < (𝑖 + 1))
108	101, 103, 105, 107	syl21anc 837	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 < (𝑖 + 1))
109	96, 97, 100, 93, 108	lttrd 11423	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < (𝑖 + 1))
110	96, 109	ltned 11398	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ (𝑖 + 1))
111	110	necomd 2995	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ≠ 𝑗)
112	111	neneqd 2944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ (𝑖 + 1) = 𝑗)
113	96, 97, 98, 93, 105	ltletrd 11422	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝑖)
114	96, 113	ltned 11398	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ 𝑖)
115	114	necomd 2995	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
116	115	neneqd 2944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
117	112, 116	2falsed 376	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = 𝑗 ↔ 𝑖 = 𝑗))
118	95, 117	bitrd 279	. . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
120	119	iffalsed 4535	. . . . . . . . . . 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 12283	. . . . . . . . . . . . 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 12283	. . . . . . . . . . . . 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 11257	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 1 ∈ ℂ)
127		simpr 484	. . . . . . . . . . . 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 11468	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 = 𝑗)
129		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
130	129	oveq1d 7447	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → (𝑖 + 1) = (𝑗 + 1))
131	128, 130	impbida 800	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = (𝑗 + 1) ↔ 𝑖 = 𝑗))
132	121, 131	bitrd 279	. . . . . . . . 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 812	. . . . . . . 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 279	. . . . . . 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 812	. . . . . 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 4548	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1r‘𝑅), (0g‘𝑅)) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
137		eqid 2736	. . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
138		fzfid 14015	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ∈ Fin)
139		eqid 2736	. . . . . 6 ⊢ (1r‘((1...(𝑁 − 1)) Mat 𝑅)) = (1r‘((1...(𝑁 − 1)) Mat 𝑅))
140	137, 26, 27, 138, 29, 19, 21, 139	mat1ov 22455	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
141	136, 140	eqtr4d 2779	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1r‘𝑅), (0g‘𝑅)) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
142	25, 51, 141	3eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
143	142	ralrimivva 3201	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
144	1, 2, 2, 4, 4, 16	smatrcl 33796	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
145		elmapfn 8906	. . . 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 14014	. . . . . 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 22456	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1...(𝑁 − 1)) ∈ Fin) → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
150	6, 147, 149	sylancl 586	. . . . 5 ⊢ (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
151	137, 13	matbas2 22428	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
153	150, 152	eleqtrrd 2843	. . . 4 ⊢ (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
154		elmapfn 8906	. . . 4 ⊢ ((1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
155	153, 154	syl 17	. . 3 ⊢ (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
156		eqfnov2 7564	. . 3 ⊢ (((𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))) ∧ (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → ((𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
157	146, 155, 156	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
158	143, 157	mpbird 257	1 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ifcif 4524   class class class wbr 5142   × cxp 5682   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  Fincfn 8986  ℂcc 11154  ℝcr 11155  1c1 11157   + caddc 11159   < clt 11296   ≤ cle 11297   − cmin 11493  ℕcn 12267  ℤcz 12615  ℤ_≥cuz 12879  ...cfz 13548  Basecbs 17248  0_gc0g 17485  1_rcur 20179  Ringcrg 20231   Mat cmat 22412  subMat1csmat 33793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-subrg 20571  df-lmod 20861  df-lss 20931  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-mamu 22396  df-mat 22413  df-smat 33794

This theorem is referenced by: (None)