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Theorem scmatscmiddistr 21755

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 768	. . . . . 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 21742	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ (𝑁 DMat 𝑅))
8	2, 7	eqeltrid 2841	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (𝑁 DMat 𝑅))
9	8	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (𝑁 DMat 𝑅))
10	1, 9	jca 512	. . . . 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 21750	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
14	10, 13	syldan 591	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
15		simprr 770	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑇 ∈ 𝐵)
16	15, 9	jca 512	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅)))
17	11, 3, 4, 12, 6	dmatscmcl 21750	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
18	16, 17	syldan 591	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
19	14, 18	jca 512	. . 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 7342	. . . 4 ⊢ ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 ∗ 1 )(._r‘𝐴)(𝑇 ∗ 1 ))
22	3, 4, 5, 6	dmatmul 21744	. . . 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 2788	. . 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 591	. 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 764	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑁 ∈ Fin)
26		simplr 766	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑅 ∈ Ring)
27	25, 26, 1	3jca 1127	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
28	27	3ad2ant1 1132	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
29		3simpc 1149	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
30	3, 11, 5, 2, 12	scmatscmide 21754	. . . . . . 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 584	. . . . . 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 1127	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵))
33	32	3ad2ant1 1132	. . . . . . 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 21754	. . . . . . 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 584	. . . . . 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 7347	. . . . 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 4491	. . . 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 7406	. . 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 4478	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆)
40		iftrue 4478	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇)
41	39, 40	oveq12d 7347	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(._r‘𝑅)𝑇))
42	41	adantl 482	. . . . . 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 4503	. . . . 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 7406	. . . 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 7342	. . . . . . . . . . 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 4496	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
52	51	adantl 482	. . . . . . . 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 768	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
54		simprr 770	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
55		ovex 7362	. . . . . . . . . 10 ⊢ (𝑆 · 𝑇) ∈ V
56	5	fvexi 6833	. . . . . . . . . 10 ⊢ 0 ∈ V
57	55, 56	ifex 4522	. . . . . . . . 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 7479	. . . . . . 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 1127	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵))
61	11, 47	ringcl 19887	. . . . . . . . . 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 1127	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵))
64	3, 11, 5, 2, 12	scmatscmide 21754	. . . . . . . 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 580	. . . . . . 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 2779	. . . . . 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 3193	. . . . 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 19887	. . . . . . . . . 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 19895	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
72	71	adantl 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐵)
73	72	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 0 ∈ 𝐵)
74	70, 73	ifcld 4518	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) ∈ 𝐵)
75	74	3ad2ant1 1132	. . . . . . 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 21670	. . . . . 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 21690	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
78	4, 2	ringidcl 19894	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
79	77, 78	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
80	79	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (Base‘𝐴))
81	62, 80	jca 512	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴)))
82	11, 3, 4, 12	matvscl 21678	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
83	81, 82	syldan 591	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
84	3, 4	eqmat 21671	. . . . . 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 584	. . . . 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 256	. . . 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 2776	. . 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 2776	. 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 2776	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3061 Vcvv 3441 ifcif 4472 ‘cfv 6473 (class class class)co 7329 ∈ cmpo 7331 Fincfn 8796 Basecbs 17001 ._rcmulr 17052 ·_𝑠 cvsca 17055 0_gc0g 17239 1_rcur 19824 Ringcrg 19870 Mat cmat 21652 DMat cdmat 21735

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-fzo 13476 df-seq 13815 df-hash 14138 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-hom 17075 df-cco 17076 df-0g 17241 df-gsum 17242 df-prds 17247 df-pws 17249 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-ghm 18920 df-cntz 19011 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-subrg 20119 df-lmod 20223 df-lss 20292 df-sra 20532 df-rgmod 20533 df-dsmm 21037 df-frlm 21052 df-mamu 21631 df-mat 21653 df-dmat 21737

This theorem is referenced by: scmatmulcl 21765 scmatmhm 21781

W3C validator