Theorem 1marepvmarrepid 22769

Description: Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)

Hypotheses

Ref	Expression
marepvmarrep1.v	⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
marepvmarrep1.o	⊢ 1 = (1r‘(𝑁 Mat 𝑅))
marepvmarrep1.x	⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)

Assertion

Ref	Expression
1marepvmarrepid	⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = 𝑋)

Proof of Theorem 1marepvmarrepid

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvmarrep1.x	. . . 4 ⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)
2		eqid 2766	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2766	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
4		marepvmarrep1.v	. . . . . 6 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
5		marepvmarrep1.o	. . . . . 6 ⊢ 1 = (1r‘(𝑁 Mat 𝑅))
6	2, 3, 4, 5	ma1repvcl 22764	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
7	6	ancom2s 663	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
8	1, 7	eqeltrid 2870	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9		elmapi 8855	. . . . . . 7 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10		ffvelcdm 7083	. . . . . . . 8 ⊢ ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼 ∈ 𝑁) → (𝑍‘𝐼) ∈ (Base‘𝑅))
11	10	ex 418	. . . . . . 7 ⊢ (𝑍:𝑁⟶(Base‘𝑅) → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
12	9, 11	syl 18	. . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
13	12, 4	eleq2s 2884	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
14	13	impcom 413	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → (𝑍‘𝐼) ∈ (Base‘𝑅))
15	14	adantl 487	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑍‘𝐼) ∈ (Base‘𝑅))
16		simpl 488	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝐼 ∈ 𝑁)
17	16	adantl 487	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝐼 ∈ 𝑁)
18		eqid 2766	. . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19		eqid 2766	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
20	2, 3, 18, 19	marrepval 22756	. . 3 ⊢ (((𝑋 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝑍‘𝐼) ∈ (Base‘𝑅)) ∧ (𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗))))
21	8, 15, 17, 17, 20	syl22anc 852	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗))))
22		iftrue 4498	. . . . . 6 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)))
23	22	adantr 486	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)))
24		iftrue 4498	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (𝑍‘𝐼))
25	24	adantr 486	. . . . . . 7 ⊢ ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (𝑍‘𝐼))
26		iftrue 4498	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝑖))
27		fveq2 6888	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑍‘𝑖) = (𝑍‘𝐼))
28	27	adantr 486	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑍‘𝑖) = (𝑍‘𝐼))
29	26, 28	sylan9eq 2821	. . . . . . 7 ⊢ ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝐼))
30	25, 29	eqtr4d 2804	. . . . . 6 ⊢ ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
31		eqid 2766	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
32		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
33	32	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑁 ∈ Fin)
34	33	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
35		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
36	35	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑅 ∈ Ring)
37	36	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1155	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
39		simp3 1156	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
40	2, 31, 19, 34, 37, 38, 39, 5	mat1ov 22642	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
41	40	adantl 487	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
42	41	adantl 487	. . . . . . . 8 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
43		eqtr2 2787	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝐼 = 𝑗)
44	43	eqcomd 2772	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝑗 = 𝐼)
45	44	ex 418	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑗 → 𝑗 = 𝐼))
46	45	con3d 153	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
47	46	adantr 486	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
48	47	impcom 413	. . . . . . . . 9 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ¬ 𝑖 = 𝑗)
49		iffalse 4501	. . . . . . . . 9 ⊢ (¬ 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
50	48, 49	syl 18	. . . . . . . 8 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
51	42, 50	eqtrd 2801	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑖 1 𝑗) = (0g‘𝑅))
52		iffalse 4501	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
53	52	adantr 486	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
54		iffalse 4501	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (0g‘𝑅))
55	54	adantr 486	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (0g‘𝑅))
56	51, 53, 55	3eqtr4rd 2812	. . . . . 6 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
57	30, 56	pm2.61ian 824	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
58	23, 57	eqtrd 2801	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
59		iffalse 4501	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
60	59	adantr 486	. . . . 5 ⊢ ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
61	2, 3, 5	mat1bas 22643	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
62	61	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
64	63	adantl 487	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉)
65	62, 64, 17	3jca 1146	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
66	65	3ad2ant1 1151	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
67		3simpc 1168	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
68	37, 66, 67	3jca 1146	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)))
69	68	adantl 487	. . . . . 6 ⊢ ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)))
70	2, 3, 4, 5, 19, 1	ma1repveval 22765	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑋𝑗) = if(𝑗 = 𝐼, (𝑍‘𝑖), if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))))
71	69, 70	syl 18	. . . . 5 ⊢ ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑋𝑗) = if(𝑗 = 𝐼, (𝑍‘𝑖), if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))))
72	34	ad2antlr 740	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
73	37	ad2antlr 740	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
74	38	ad2antlr 740	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖 ∈ 𝑁)
75	39	ad2antlr 740	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑗 ∈ 𝑁)
76	2, 31, 19, 72, 73, 74, 75, 5	mat1ov 22642	. . . . . . 7 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
77		equcom 2051	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 ↔ 𝑗 = 𝑖)
78	77	a1i 11	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗 ↔ 𝑗 = 𝑖))
79	78	ifbid 4516	. . . . . . 7 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)))
80	76, 79	eqtr2d 2802	. . . . . 6 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅)) = (𝑖 1 𝑗))
81	80	ifeq2da 4525	. . . . 5 ⊢ ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐼, (𝑍‘𝑖), if(𝑗 = 𝑖, (1r‘𝑅), (0g‘𝑅))) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
82	60, 71, 81	3eqtrd 2805	. . . 4 ⊢ ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
83	58, 82	pm2.61ian 824	. . 3 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)))
84	83	mpoeq3dva 7500	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))))
85		eqid 2766	. . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
86	2, 3, 85, 4	marepvval 22761	. . . 4 ⊢ (( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))))
87	65, 86	syl 18	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))))
88	1, 87	eqtr2id 2814	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))) = 𝑋)
89	21, 84, 88	3eqtrd 2805	1 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ifcif 4492  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425   ↑_m cmap 8833  Fincfn 8952  Basecbs 17294  0_gc0g 17517  1_rcur 20294  Ringcrg 20346   Mat cmat 22601   matRRep cmarrep 22750   matRepV cmatrepV 22751

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687  df-om 7872  df-1st 7995  df-2nd 7996  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-fzo 13702  df-seq 14058  df-hash 14387  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-ip 17353  df-tset 17354  df-ple 17355  df-ds 17357  df-hom 17359  df-cco 17360  df-0g 17519  df-gsum 17520  df-prds 17525  df-pws 17527  df-mre 17663  df-mrc 17664  df-acs 17666  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-submnd 18873  df-grp 19034  df-minusg 19035  df-sbg 19036  df-mulg 19165  df-subg 19220  df-ghm 19315  df-cntz 19418  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-subrg 20706  df-lmod 21020  df-lss 21090  df-sra 21331  df-rgmod 21332  df-dsmm 21919  df-frlm 21934  df-mamu 22585  df-mat 22602  df-marrep 22752  df-marepv 22753

This theorem is used by:  cramerimplem1  22877