1smat1 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1smat1		Structured version Visualization version GIF version

Theorem 1smat1 31656

Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 21640. (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 2738	. . . . 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 13620	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
8		eqid 2738	. . . . . . . . 9 ⊢ ((1...𝑁) Mat 𝑅) = ((1...𝑁) Mat 𝑅)
9		eqid 2738	. . . . . . . . 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 21506	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1...𝑁) ∈ Fin) → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
12	6, 7, 11	sylancl 585	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
13		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	8, 13	matbas2 21478	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
15	7, 6, 14	sylancr 586	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
16	12, 15	eleqtrrd 2842	. . . . . 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 13216	. . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
19		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
20	18, 19	sselid 3915	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
21		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
22	18, 21	sselid 3915	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
23		eqidd 2739	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
24		eqidd 2739	. . . . 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 31649	. . . 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 2738	. . . . 5 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
27		eqid 2738	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑅 ∈ Ring)
30		nnuz 12550	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
31	20, 30	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (ℤ_≥‘1))
32		fznatpl1 13239	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → (𝑖 + 1) ∈ (1...𝑁))
33	3, 19, 32	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 + 1) ∈ (1...𝑁))
34		peano2fzr 13198	. . . . . . . 8 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ (𝑖 + 1) ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
35	31, 33, 34	syl2anc 583	. . . . . . 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 2826	. . . . . . 7 ⊢ (𝑖 = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → (𝑖 ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
38		eleq1 2826	. . . . . . 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 4495	. . . . . 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 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (ℤ_≥‘1))
42		fznatpl1 13239	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝑗 + 1) ∈ (1...𝑁))
43	3, 21, 42	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 + 1) ∈ (1...𝑁))
44		peano2fzr 13198	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘1) ∧ (𝑗 + 1) ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
45	41, 43, 44	syl2anc 583	. . . . . . 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 2826	. . . . . . 7 ⊢ (𝑗 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → (𝑗 ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
48		eleq1 2826	. . . . . . 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 4495	. . . . . 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 21505	. . . 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 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → 𝑖 < 𝐼)
53	52	iftrued 4464	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = 𝑖)
54	53	eqeq1d 2740	. . . . . . . 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 4464	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
57	56	eqeq2d 2749	. . . . . . . . 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 4467	. . . . . . . . . . 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 2749	. . . . . . . . . 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 11918	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℝ)
62	61	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
63		fz1ssnn 13216	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ⊆ ℕ
64	63, 4	sselid 3915	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℕ)
65	64	nnred 11918	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ)
66	65	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
67	22	nnred 11918	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℝ)
68	67	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
69		1red 10907	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 1 ∈ ℝ)
70	68, 69	readdcld 10935	. . . . . . . . . . . . . 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 12354	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ ℤ)
73	72	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
74	22	nnzd 12354	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℤ)
75	74	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℤ)
76	66, 68, 58	nltled 11055	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ≤ 𝑗)
77		zleltp1 12301	. . . . . . . . . . . . . . . 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 834	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 < (𝑗 + 1))
80	62, 66, 70, 71, 79	lttrd 11066	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < (𝑗 + 1))
81	62, 80	ltned 11041	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ (𝑗 + 1))
82	81	neneqd 2947	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = (𝑗 + 1))
83	62, 66, 68, 71, 76	ltletrd 11065	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝑗)
84	62, 83	ltned 11041	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
85	84	neneqd 2947	. . . . . . . . . . 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 278	. . . . . . . . 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 809	. . . . . . . 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 278	. . . . . . 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 4467	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = (𝑖 + 1))
92	91	eqeq1d 2740	. . . . . . . 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 4464	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
95	94	eqeq2d 2749	. . . . . . . . . 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 722	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
97	65	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
98	61	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
99		1red 10907	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 1 ∈ ℝ)
100	98, 99	readdcld 10935	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ∈ ℝ)
101	72	ad3antrrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
102	20	nnzd 12354	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℤ)
103	102	ad2antrr 722	. . . . . . . . . . . . . . . 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 11055	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ≤ 𝑖)
106		zleltp1 12301	. . . . . . . . . . . . . . . . 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 834	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 < (𝑖 + 1))
109	96, 97, 100, 93, 108	lttrd 11066	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < (𝑖 + 1))
110	96, 109	ltned 11041	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ (𝑖 + 1))
111	110	necomd 2998	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ≠ 𝑗)
112	111	neneqd 2947	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ (𝑖 + 1) = 𝑗)
113	96, 97, 98, 93, 105	ltletrd 11065	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝑖)
114	96, 113	ltned 11041	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ 𝑖)
115	114	necomd 2998	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ≠ 𝑗)
116	115	neneqd 2947	. . . . . . . . . . 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 278	. . . . . . . . 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 4467	. . . . . . . . . . 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 2749	. . . . . . . . . 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 11919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℂ)
123	122	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 ∈ ℂ)
124	22	nncnd 11919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℂ)
125	124	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑗 ∈ ℂ)
126		1cnd 10901	. . . . . . . . . . . 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 11111	. . . . . . . . . . 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 7270	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → (𝑖 + 1) = (𝑗 + 1))
131	128, 130	impbida 797	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = (𝑗 + 1) ↔ 𝑖 = 𝑗))
132	121, 131	bitrd 278	. . . . . . . . 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 809	. . . . . . . 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 278	. . . . . . 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 809	. . . . . 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 4479	. . . . 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 2738	. . . . . 6 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
138		fzfid 13621	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ∈ Fin)
139		eqid 2738	. . . . . 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 21505	. . . . 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 2781	. . . 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 2782	. . 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 3114	. 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 31648	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
145		elmapfn 8611	. . . 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 13620	. . . . . 6 ⊢ (1...(𝑁 − 1)) ∈ Fin
148		eqid 2738	. . . . . . 7 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	137, 148, 139	mat1bas 21506	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
151	137, 13	matbas2 21478	. . . . . 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 586	. . . . 5 ⊢ (𝜑 → ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
153	150, 152	eleqtrrd 2842	. . . 4 ⊢ (𝜑 → (1_r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑_m ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
154		elmapfn 8611	. . . 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 7382	. . 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 583	. 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 256	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 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ifcif 4456 class class class wbr 5070 × cxp 5578 Fn wfn 6413 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 Basecbs 16840 0_gc0g 17067 1_rcur 19652 Ringcrg 19698 Mat cmat 21464 subMat1csmat 31645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-dsmm 20849 df-frlm 20864 df-mamu 21443 df-mat 21465 df-smat 31646

This theorem is referenced by: (None)

W3C validator