Theorem 1marepvmarrepid 20882

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 2772	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2772	. . . . . 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 20877	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
7	6	ancom2s 637	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
8	1, 7	syl5eqel 2864	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9		elmapi 8222	. . . . . . 7 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10		ffvelrn 6668	. . . . . . . 8 ⊢ ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼 ∈ 𝑁) → (𝑍‘𝐼) ∈ (Base‘𝑅))
11	10	ex 405	. . . . . . 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 2878	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐼 ∈ 𝑁 → (𝑍‘𝐼) ∈ (Base‘𝑅)))
14	13	impcom 399	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → (𝑍‘𝐼) ∈ (Base‘𝑅))
15	14	adantl 474	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑍‘𝐼) ∈ (Base‘𝑅))
16		simpl 475	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝐼 ∈ 𝑁)
17	16	adantl 474	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝐼 ∈ 𝑁)
18		eqid 2772	. . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19		eqid 2772	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
20	2, 3, 18, 19	marrepval 20869	. . 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 826	. 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 4350	. . . . . 6 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)))
23	22	adantr 473	. . . . 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 4350	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (𝑍‘𝐼))
25	24	adantr 473	. . . . . . 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 4350	. . . . . . . 8 ⊢ (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑍‘𝑖))
27		fveq2 6493	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑍‘𝑖) = (𝑍‘𝐼))
28	27	adantr 473	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑍‘𝑖) = (𝑍‘𝐼))
29	26, 28	sylan9eq 2828	. . . . . . 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 2811	. . . . . 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 2772	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
32		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
33	32	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑁 ∈ Fin)
34	33	3ad2ant1 1113	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
35		simpl 475	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
36	35	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑅 ∈ Ring)
37	36	3ad2ant1 1113	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1117	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
39		simp3 1118	. . . . . . . . . . 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 20755	. . . . . . . . . 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 474	. . . . . . . . 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 474	. . . . . . . 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 2794	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝐼 = 𝑗)
44	43	eqcomd 2778	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑖 = 𝑗) → 𝑗 = 𝐼)
45	44	ex 405	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑗 → 𝑗 = 𝐼))
46	45	con3d 150	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
47	46	adantr 473	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
48	47	impcom 399	. . . . . . . . 9 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ¬ 𝑖 = 𝑗)
49		iffalse 4353	. . . . . . . . 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 2808	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑖 1 𝑗) = (0g‘𝑅))
52		iffalse 4353	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
53	52	adantr 473	. . . . . . 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 4353	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)) = (0g‘𝑅))
55	54	adantr 473	. . . . . . 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 2819	. . . . . 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 799	. . . . 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 2808	. . . 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 4353	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍‘𝐼), (0g‘𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
60	59	adantr 473	. . . . 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 20756	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
62	61	adantr 473	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63		simpr 477	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
64	63	adantl 474	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉)
65	62, 64, 17	3jca 1108	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
66	65	3ad2ant1 1113	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁))
67		3simpc 1130	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
68	37, 66, 67	3jca 1108	. . . . . . 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 474	. . . . . 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 20878	. . . . . 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 714	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
73	37	ad2antlr 714	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
74	38	ad2antlr 714	. . . . . . . 8 ⊢ (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖 ∈ 𝑁)
75	39	ad2antlr 714	. . . . . . . 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 20755	. . . . . . 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 1975	. . . . . . . . 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 4366	. . . . . . 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 2809	. . . . . 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 4375	. . . . 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 2812	. . . 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 799	. . 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 7043	. 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 2772	. . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
86	2, 3, 85, 4	marepvval 20874	. . . 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	syl5req 2821	. 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 2812	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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ifcif 4344  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972   ↑_𝑚 cmap 8200  Fincfn 8300  Basecbs 16333  0_gc0g 16563  1_rcur 18968  Ringcrg 19014   Mat cmat 20714   matRRep cmarrep 20863   matRepV cmatrepV 20864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-ot 4444  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7495  df-2nd 7496  df-supp 7628  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-ixp 8254  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-fsupp 8623  df-sup 8695  df-oi 8763  df-card 9156  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-3 11498  df-4 11499  df-5 11500  df-6 11501  df-7 11502  df-8 11503  df-9 11504  df-n0 11702  df-z 11788  df-dec 11906  df-uz 12053  df-fz 12703  df-fzo 12844  df-seq 13179  df-hash 13500  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-ress 16341  df-plusg 16428  df-mulr 16429  df-sca 16431  df-vsca 16432  df-ip 16433  df-tset 16434  df-ple 16435  df-ds 16437  df-hom 16439  df-cco 16440  df-0g 16565  df-gsum 16566  df-prds 16571  df-pws 16573  df-mre 16709  df-mrc 16710  df-acs 16712  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-mhm 17797  df-submnd 17798  df-grp 17888  df-minusg 17889  df-sbg 17890  df-mulg 18006  df-subg 18054  df-ghm 18121  df-cntz 18212  df-cmn 18662  df-abl 18663  df-mgp 18957  df-ur 18969  df-ring 19016  df-subrg 19250  df-lmod 19352  df-lss 19420  df-sra 19660  df-rgmod 19661  df-dsmm 20572  df-frlm 20587  df-mamu 20691  df-mat 20715  df-marrep 20865  df-marepv 20866

This theorem is referenced by:  cramerimplem1  20990