Theorem 1smat1 29870

Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20389. (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 2622	. . . . 5 ⊢ (𝐼(subMat1‘ 1 )𝐼) = (𝐼(subMat1‘ 1 )𝐼)
2		1smat1.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
4		1smat1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
6		1smat1.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		fzfi 12771	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
8		eqid 2622	. . . . . . . . 9 ⊢ ((1...𝑁) Mat 𝑅) = ((1...𝑁) Mat 𝑅)
9		eqid 2622	. . . . . . . . 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 20255	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1...𝑁) ∈ Fin) → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
12	6, 7, 11	sylancl 694	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
13		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	8, 13	matbas2 20227	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
15	7, 6, 14	sylancr 695	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
16	12, 15	eleqtrrd 2704	. . . . . 6 ⊢ (𝜑 → 1 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
17	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 1 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
18		fz1ssnn 12372	. . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
19		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
20	18, 19	sseldi 3601	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
21		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
22	18, 21	sseldi 3601	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
23		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
24		eqidd 2623	. . . . 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 29863	. . . 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 2622	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
27		eqid 2622	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑅 ∈ Ring)
30		nnuz 11723	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
31	20, 30	syl6eleq 2711	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (ℤ≥‘1))
32		fznatpl1 12395	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → (𝑖 + 1) ∈ (1...𝑁))
33	3, 19, 32	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 + 1) ∈ (1...𝑁))
34		peano2fzr 12354	. . . . . . . 8 ⊢ ((𝑖 ∈ (ℤ≥‘1) ∧ (𝑖 + 1) ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
35	31, 33, 34	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
36	35, 33	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)))
37		eleq1 2689	. . . . . . 7 ⊢ (𝑖 = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → (𝑖 ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
38		eleq1 2689	. . . . . . 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 4124	. . . . . 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	syl6eleq 2711	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (ℤ≥‘1))
42		fznatpl1 12395	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝑗 + 1) ∈ (1...𝑁))
43	3, 21, 42	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 + 1) ∈ (1...𝑁))
44		peano2fzr 12354	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ≥‘1) ∧ (𝑗 + 1) ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
45	41, 43, 44	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
46	45, 43	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)))
47		eleq1 2689	. . . . . . 7 ⊢ (𝑗 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → (𝑗 ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
48		eleq1 2689	. . . . . . 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 4124	. . . . . 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 20254	. . . 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 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → 𝑖 < 𝐼)
53	52	iftrued 4094	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = 𝑖)
54	53	eqeq1d 2624	. . . . . . . 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 477	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
56	55	iftrued 4094	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
57	56	eqeq2d 2632	. . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
59	58	iffalsed 4097	. . . . . . . . . . 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 2632	. . . . . . . . . 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 11035	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℝ)
62	61	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
63		fz1ssnn 12372	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ⊆ ℕ
64	63, 4	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
65	64	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
66	65	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
67	22	nnred 11035	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℝ)
68	67	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
69		1red 10055	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 1 ∈ ℝ)
70	68, 69	readdcld 10069	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑗 + 1) ∈ ℝ)
71	52	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝐼)
72	64	nnzd 11481	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℤ)
73	72	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
74	22	nnzd 11481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℤ)
75	74	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℤ)
76	66, 68, 58	nltled 10187	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ≤ 𝑗)
77		zleltp1 11428	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐼 ≤ 𝑗 ↔ 𝐼 < (𝑗 + 1)))
78	77	biimpa 501	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ 𝐼 ≤ 𝑗) → 𝐼 < (𝑗 + 1))
79	73, 75, 76, 78	syl21anc 1325	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 < (𝑗 + 1))
80	62, 66, 70, 71, 79	lttrd 10198	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < (𝑗 + 1))
81	62, 80	ltned 10173	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ (𝑗 + 1))
82	81	neneqd 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = (𝑗 + 1))
83	62, 66, 68, 71, 76	ltletrd 10197	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝑗)
84	62, 83	ltned 10173	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
85	84	neneqd 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
86	82, 85	2falsed 366	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = (𝑗 + 1) ↔ 𝑖 = 𝑗))
87	60, 86	bitrd 268	. . . . . . . . 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 832	. . . . . . . 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 268	. . . . . . 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 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ¬ 𝑖 < 𝐼)
91	90	iffalsed 4097	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = (𝑖 + 1))
92	91	eqeq1d 2624	. . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
94	93	iftrued 4094	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
95	94	eqeq2d 2632	. . . . . . . . . 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 762	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
97	65	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
98	61	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
99		1red 10055	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 1 ∈ ℝ)
100	98, 99	readdcld 10069	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ∈ ℝ)
101	72	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
102	20	nnzd 11481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℤ)
103	102	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℤ)
104	90	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 < 𝐼)
105	97, 98, 104	nltled 10187	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ≤ 𝑖)
106		zleltp1 11428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝐼 ≤ 𝑖 ↔ 𝐼 < (𝑖 + 1)))
107	106	biimpa 501	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝐼 ≤ 𝑖) → 𝐼 < (𝑖 + 1))
108	101, 103, 105, 107	syl21anc 1325	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 < (𝑖 + 1))
109	96, 97, 100, 93, 108	lttrd 10198	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < (𝑖 + 1))
110	96, 109	ltned 10173	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ (𝑖 + 1))
111	110	necomd 2849	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ≠ 𝑗)
112	111	neneqd 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ (𝑖 + 1) = 𝑗)
113	96, 97, 98, 93, 105	ltletrd 10197	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝑖)
114	96, 113	ltned 10173	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ 𝑖)
115	114	necomd 2849	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
116	115	neneqd 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
117	112, 116	2falsed 366	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = 𝑗 ↔ 𝑖 = 𝑗))
118	95, 117	bitrd 268	. . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
120	119	iffalsed 4097	. . . . . . . . . . 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 2632	. . . . . . . . . 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 11036	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℂ)
123	122	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 ∈ ℂ)
124	22	nncnd 11036	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℂ)
125	124	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑗 ∈ ℂ)
126		1cnd 10056	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 1 ∈ ℂ)
127		simpr 477	. . . . . . . . . . . 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 10242	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 = 𝑗)
129		simpr 477	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
130	129	oveq1d 6665	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → (𝑖 + 1) = (𝑗 + 1))
131	128, 130	impbida 877	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = (𝑗 + 1) ↔ 𝑖 = 𝑗))
132	121, 131	bitrd 268	. . . . . . . . 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 832	. . . . . . . 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 268	. . . . . . 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 832	. . . . . 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 4108	. . . . 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 2622	. . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
138		fzfid 12772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ∈ Fin)
139		eqid 2622	. . . . . 6 ⊢ (1r‘((1...(𝑁 − 1)) Mat 𝑅)) = (1r‘((1...(𝑁 − 1)) Mat 𝑅))
140	137, 26, 27, 138, 29, 19, 21, 139	mat1ov 20254	. . . . 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 2659	. . . 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 2660	. . 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 2971	. 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 29862	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
145		elmapfn 7880	. . . 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 12771	. . . . . 6 ⊢ (1...(𝑁 − 1)) ∈ Fin
148		eqid 2622	. . . . . . 7 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	137, 148, 139	mat1bas 20255	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1...(𝑁 − 1)) ∈ Fin) → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
150	6, 147, 149	sylancl 694	. . . . 5 ⊢ (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
151	137, 13	matbas2 20227	. . . . . 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 695	. . . . 5 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
153	150, 152	eleqtrrd 2704	. . . 4 ⊢ (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
154		elmapfn 7880	. . . 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 6767	. . 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 693	. 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 247	1 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ifcif 4086   class class class wbr 4653   × cxp 5112   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  ℂcc 9934  ℝcr 9935  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  Basecbs 15857  0_gc0g 16100  1_rcur 18501  Ringcrg 18547   Mat cmat 20213  subMat1csmat 29859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-smat 29860

This theorem is referenced by: (None)