Theorem 1marepvmarrepid 21444

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 2734	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2734	. . . . . 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 21439	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
7	6	ancom2s 650	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
8	1, 7	eqeltrid 2838	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9		elmapi 8519	. . . . . . 7 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10		ffvelrn 6891	. . . . . . . 8 ⊢ ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼 ∈ 𝑁) → (𝑍‘𝐼) ∈ (Base‘𝑅))
11	10	ex 416	. . . . . . 7 ⊢ (𝑍:𝑁⟶(Base‘𝑅) → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
12	9, 11	syl 17	. . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
13	12, 4	eleq2s 2852	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
14	13	impcom 411	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → (𝑍‘𝐼) ∈ (Base‘𝑅))
15	14	adantl 485	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑍‘𝐼) ∈ (Base‘𝑅))
16		simpl 486	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝐼 ∈ 𝑁)
17	16	adantl 485	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝐼 ∈ 𝑁)
18		eqid 2734	. . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19		eqid 2734	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
20	2, 3, 18, 19	marrepval 21431	. . 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 839	. 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 4435	. . . . . 6 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)))
23	22	adantr 484	. . . . 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 4435	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (𝑍‘𝐼))
25	24	adantr 484	. . . . . . 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 4435	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝑖))
27		fveq2 6706	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑍‘𝑖) = (𝑍‘𝐼))
28	27	adantr 484	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑍‘𝑖) = (𝑍‘𝐼))
29	26, 28	sylan9eq 2794	. . . . . . 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 2777	. . . . . 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 2734	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
32		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
33	32	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑁 ∈ Fin)
34	33	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
35		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
36	35	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑅 ∈ Ring)
37	36	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1139	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
39		simp3 1140	. . . . . . . . . . 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 21317	. . . . . . . . . 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 485	. . . . . . . . 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 485	. . . . . . . 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 2758	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝐼 = 𝑗)
44	43	eqcomd 2740	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝑗 = 𝐼)
45	44	ex 416	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑗 → 𝑗 = 𝐼))
46	45	con3d 155	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
47	46	adantr 484	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
48	47	impcom 411	. . . . . . . . 9 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ¬ 𝑖 = 𝑗)
49		iffalse 4438	. . . . . . . . 9 ⊢ (¬ 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
50	48, 49	syl 17	. . . . . . . 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 2774	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑖 1 𝑗) = (0g‘𝑅))
52		iffalse 4438	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
53	52	adantr 484	. . . . . . 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 4438	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (0g‘𝑅))
55	54	adantr 484	. . . . . . 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 2785	. . . . . 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 812	. . . . 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 2774	. . . 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 4438	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
60	59	adantr 484	. . . . 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 21318	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
62	61	adantr 484	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63		simpr 488	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
64	63	adantl 485	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉)
65	62, 64, 17	3jca 1130	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
66	65	3ad2ant1 1135	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
67		3simpc 1152	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
68	37, 66, 67	3jca 1130	. . . . . . 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 485	. . . . . 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 21440	. . . . . 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 17	. . . . 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 727	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
73	37	ad2antlr 727	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
74	38	ad2antlr 727	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖 ∈ 𝑁)
75	39	ad2antlr 727	. . . . . . . 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 21317	. . . . . . 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 2026	. . . . . . . . 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 4452	. . . . . . 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 2775	. . . . . 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 4461	. . . . 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 2778	. . . 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 812	. . 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 7277	. 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 2734	. . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
86	2, 3, 85, 4	marepvval 21436	. . . 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 17	. . 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 2787	. 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 2778	1 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ifcif 4429  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202   ∈ cmpo 7204   ↑_m cmap 8497  Fincfn 8615  Basecbs 16684  0_gc0g 16916  1_rcur 19488  Ringcrg 19534   Mat cmat 21276   matRRep cmarrep 21425   matRepV cmatrepV 21426

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 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

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 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-ot 4540  df-uni 4810  df-int 4850  df-iun 4896  df-iin 4897  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-om 7634  df-1st 7750  df-2nd 7751  df-supp 7893  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-map 8499  df-ixp 8568  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fsupp 8975  df-sup 9047  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-fz 13079  df-fzo 13222  df-seq 13558  df-hash 13880  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-ress 16692  df-plusg 16780  df-mulr 16781  df-sca 16783  df-vsca 16784  df-ip 16785  df-tset 16786  df-ple 16787  df-ds 16789  df-hom 16791  df-cco 16792  df-0g 16918  df-gsum 16919  df-prds 16924  df-pws 16926  df-mre 17061  df-mrc 17062  df-acs 17064  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-mhm 18190  df-submnd 18191  df-grp 18340  df-minusg 18341  df-sbg 18342  df-mulg 18461  df-subg 18512  df-ghm 18592  df-cntz 18683  df-cmn 19144  df-abl 19145  df-mgp 19477  df-ur 19489  df-ring 19536  df-subrg 19770  df-lmod 19873  df-lss 19941  df-sra 20181  df-rgmod 20182  df-dsmm 20666  df-frlm 20681  df-mamu 21255  df-mat 21277  df-marrep 21427  df-marepv 21428

This theorem is referenced by:  cramerimplem1  21552