Theorem 1marepvmarrepid 22519

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 22514	. . . . 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 2840	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9		elmapi 8786	. . . . . . 7 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10		ffvelcdm 7026	. . . . . . . 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 22506	. . 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 838	. 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 4485	. . . . . 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 4485	. . . . . . . 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 4485	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝑖))
27		fveq2 6834	. . . . . . . . 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 1133	. . . . . . . . . . 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 1133	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1137	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
39		simp3 1138	. . . . . . . . . . 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 22392	. . . . . . . . . 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 4488	. . . . . . . . 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 4488	. . . . . . . 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 4488	. . . . . . . 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 811	. . . . 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 4488	. . . . . 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 22393	. . . . . . . . . . 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 1128	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
66	65	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
67		3simpc 1150	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
68	37, 66, 67	3jca 1128	. . . . . . 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 22515	. . . . . 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 22392	. . . . . . 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 2019	. . . . . . . . 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 4503	. . . . . . 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 4512	. . . . 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 811	. . 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 7435	. 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 22511	. . . 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 1086   = wceq 1541   ∈ wcel 2113  ifcif 4479  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ↑_m cmap 8763  Fincfn 8883  Basecbs 17136  0_gc0g 17359  1_rcur 20116  Ringcrg 20168   Mat cmat 22351   matRRep cmarrep 22500   matRepV cmatrepV 22501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-subrg 20503  df-lmod 20813  df-lss 20883  df-sra 21125  df-rgmod 21126  df-dsmm 21687  df-frlm 21702  df-mamu 22335  df-mat 22352  df-marrep 22502  df-marepv 22503

This theorem is referenced by:  cramerimplem1  22627