Theorem 1marepvmarrepid 22540

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 2736	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2736	. . . . . 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 22535	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
7	6	ancom2s 651	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
8	1, 7	eqeltrid 2840	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9		elmapi 8796	. . . . . . 7 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10		ffvelcdm 7033	. . . . . . . 8 ⊢ ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼 ∈ 𝑁) → (𝑍‘𝐼) ∈ (Base‘𝑅))
11	10	ex 412	. . . . . . 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 2854	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
14	13	impcom 407	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → (𝑍‘𝐼) ∈ (Base‘𝑅))
15	14	adantl 481	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑍‘𝐼) ∈ (Base‘𝑅))
16		simpl 482	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝐼 ∈ 𝑁)
17	16	adantl 481	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝐼 ∈ 𝑁)
18		eqid 2736	. . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19		eqid 2736	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
20	2, 3, 18, 19	marrepval 22527	. . 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 4472	. . . . . 6 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)))
23	22	adantr 480	. . . . 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 4472	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (𝑍‘𝐼))
25	24	adantr 480	. . . . . . 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 4472	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝑖))
27		fveq2 6840	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑍‘𝑖) = (𝑍‘𝐼))
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑍‘𝑖) = (𝑍‘𝐼))
29	26, 28	sylan9eq 2791	. . . . . . 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 2774	. . . . . 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 2736	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
32		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑁 ∈ Fin)
34	33	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
35		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑅 ∈ Ring)
37	36	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1138	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
39		simp3 1139	. . . . . . . . . . 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 22413	. . . . . . . . . 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 481	. . . . . . . . 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 481	. . . . . . . 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 2757	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝐼 = 𝑗)
44	43	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝑗 = 𝐼)
45	44	ex 412	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑗 → 𝑗 = 𝐼))
46	45	con3d 152	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
48	47	impcom 407	. . . . . . . . 9 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ¬ 𝑖 = 𝑗)
49		iffalse 4475	. . . . . . . . 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 2771	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑖 1 𝑗) = (0g‘𝑅))
52		iffalse 4475	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
53	52	adantr 480	. . . . . . 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 4475	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (0g‘𝑅))
55	54	adantr 480	. . . . . . 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 2782	. . . . . 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 2771	. . . 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 4475	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
60	59	adantr 480	. . . . 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 22414	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
62	61	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
64	63	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉)
65	62, 64, 17	3jca 1129	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
66	65	3ad2ant1 1134	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
67		3simpc 1151	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
68	37, 66, 67	3jca 1129	. . . . . . 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 481	. . . . . 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 22536	. . . . . 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 728	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
73	37	ad2antlr 728	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
74	38	ad2antlr 728	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖 ∈ 𝑁)
75	39	ad2antlr 728	. . . . . . . 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 22413	. . . . . . 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 2020	. . . . . . . . 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 4490	. . . . . . 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 2772	. . . . . 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 4499	. . . . 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 2775	. . . 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 7444	. 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 2736	. . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
86	2, 3, 85, 4	marepvval 22532	. . . 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 2784	. 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 2775	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ifcif 4466  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ↑_m cmap 8773  Fincfn 8893  Basecbs 17179  0_gc0g 17402  1_rcur 20162  Ringcrg 20214   Mat cmat 22372   matRRep cmarrep 22521   matRepV cmatrepV 22522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-subrg 20547  df-lmod 20857  df-lss 20927  df-sra 21168  df-rgmod 21169  df-dsmm 21712  df-frlm 21727  df-mamu 22356  df-mat 22373  df-marrep 22523  df-marepv 22524

This theorem is referenced by:  cramerimplem1  22648