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Theorem scmatscmiddistr 22473

Description: Distributive law for scalar and ring multiplication for scalar matrices expressed as multiplications of a scalar with the identity matrix. (Contributed by AV, 19-Dec-2019.)

**Hypotheses**
Ref	Expression
scmatscmide.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatscmide.b	⊢ 𝐵 = (Base‘𝑅)
scmatscmide.0	⊢ 0 = (0_g‘𝑅)
scmatscmide.1	⊢ 1 = (1_r‘𝐴)
scmatscmide.m	⊢ ∗ = ( ·_𝑠 ‘𝐴)
scmatscmiddistr.t	⊢ · = (._r‘𝑅)
scmatscmiddistr.m	⊢ × = (._r‘𝐴)

**Assertion**
Ref	Expression
scmatscmiddistr	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 ))

Proof of Theorem scmatscmiddistr Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 771	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑆 ∈ 𝐵)
2		scmatscmide.1	. . . . . . . 8 ⊢ 1 = (1_r‘𝐴)
3		scmatscmide.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
5		scmatscmide.0	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6		eqid 2736	. . . . . . . . 9 ⊢ (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅)
7	3, 4, 5, 6	dmatid 22460	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ (𝑁 DMat 𝑅))
8	2, 7	eqeltrid 2840	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (𝑁 DMat 𝑅))
9	8	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (𝑁 DMat 𝑅))
10	1, 9	jca 511	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅)))
11		scmatscmide.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
12		scmatscmide.m	. . . . . 6 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
13	11, 3, 4, 12, 6	dmatscmcl 22468	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
14	10, 13	syldan 592	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
15		simprr 773	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑇 ∈ 𝐵)
16	15, 9	jca 511	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅)))
17	11, 3, 4, 12, 6	dmatscmcl 22468	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
18	16, 17	syldan 592	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
19	14, 18	jca 511	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅)))
20		scmatscmiddistr.m	. . . . 5 ⊢ × = (._r‘𝐴)
21	20	oveqi 7380	. . . 4 ⊢ ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 ∗ 1 )(._r‘𝐴)(𝑇 ∗ 1 ))
22	3, 4, 5, 6	dmatmul 22462	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 ∗ 1 )(._r‘𝐴)(𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )))
23	21, 22	eqtrid 2783	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )))
24	19, 23	syldan 592	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )))
25		simpll 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑁 ∈ Fin)
26		simplr 769	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑅 ∈ Ring)
27	25, 26, 1	3jca 1129	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
28	27	3ad2ant1 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
29		3simpc 1151	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
30	3, 11, 5, 2, 12	scmatscmide 22472	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑆 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 ))
31	28, 29, 30	syl2anc 585	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑆 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 ))
32	25, 26, 15	3jca 1129	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵))
33	32	3ad2ant1 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵))
34	3, 11, 5, 2, 12	scmatscmide 22472	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 ))
35	33, 29, 34	syl2anc 585	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑇 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 ))
36	31, 35	oveq12d 7385	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)) = (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )))
37	36	ifeq1d 4486	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 ) = if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ))
38	37	mpoeq3dva 7444	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )))
39		iftrue 4472	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆)
40		iftrue 4472	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇)
41	39, 40	oveq12d 7385	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(._r‘𝑅)𝑇))
42	41	adantl 481	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 = 𝑗) → (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(._r‘𝑅)𝑇))
43	42	ifeq1da 4498	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ) = if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))
44	43	mpoeq3dva 7444	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )))
45		eqidd 2737	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )))
46		eqeq12 2753	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 = 𝑗 ↔ 𝑥 = 𝑦))
47		scmatscmiddistr.t	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑅)
48	47	eqcomi 2745	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = ·
49	48	oveqi 7380	. . . . . . . . . . 11 ⊢ (𝑆(._r‘𝑅)𝑇) = (𝑆 · 𝑇)
50	49	a1i 11	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑆(._r‘𝑅)𝑇) = (𝑆 · 𝑇))
51	46, 50	ifbieq1d 4491	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
52	51	adantl 481	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑦)) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
53		simprl 771	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
54		simprr 773	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
55		ovex 7400	. . . . . . . . . 10 ⊢ (𝑆 · 𝑇) ∈ V
56	5	fvexi 6854	. . . . . . . . . 10 ⊢ 0 ∈ V
57	55, 56	ifex 4517	. . . . . . . . 9 ⊢ if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈ V
58	57	a1i 11	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈ V)
59	45, 52, 53, 54, 58	ovmpod 7519	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
60	26, 1, 15	3jca 1129	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵))
61	11, 47	ringcl 20231	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆 · 𝑇) ∈ 𝐵)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 · 𝑇) ∈ 𝐵)
63	25, 26, 62	3jca 1129	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵))
64	3, 11, 5, 2, 12	scmatscmide 22472	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
65	63, 64	sylan 581	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
66	59, 65	eqtr4d 2774	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦))
67	66	ralrimivva 3180	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦))
68		eqid 2736	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
69	11, 68	ringcl 20231	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆(._r‘𝑅)𝑇) ∈ 𝐵)
70	60, 69	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆(._r‘𝑅)𝑇) ∈ 𝐵)
71	11, 5	ring0cl 20248	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
72	71	adantl 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐵)
73	72	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 0 ∈ 𝐵)
74	70, 73	ifcld 4513	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) ∈ 𝐵)
75	74	3ad2ant1 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) ∈ 𝐵)
76	3, 11, 4, 25, 26, 75	matbas2d 22388	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) ∈ (Base‘𝐴))
77	3	matring 22408	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
78	4, 2	ringidcl 20246	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
79	77, 78	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
80	79	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (Base‘𝐴))
81	62, 80	jca 511	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴)))
82	11, 3, 4, 12	matvscl 22396	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
83	81, 82	syldan 592	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
84	3, 4	eqmat 22389	. . . . . 6 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) ∈ (Base‘𝐴) ∧ ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦)))
85	76, 83, 84	syl2anc 585	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦)))
86	67, 85	mpbird 257	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 ))
87	44, 86	eqtrd 2771	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = ((𝑆 · 𝑇) ∗ 1 ))
88	38, 87	eqtrd 2771	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(._r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )) = ((𝑆 · 𝑇) ∗ 1 ))
89	24, 88	eqtrd 2771	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 ifcif 4466 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 Fincfn 8893 Basecbs 17179 ._rcmulr 17221 ·_𝑠 cvsca 17224 0_gc0g 17402 1_rcur 20162 Ringcrg 20214 Mat cmat 22372 DMat cdmat 22453

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-dmat 22455

This theorem is referenced by: scmatmulcl 22483 scmatmhm 22499

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