scmatscmiddistr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > scmatscmiddistr		Structured version Visualization version GIF version

Theorem scmatscmiddistr 22535

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 770	. . . . . 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 2740	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
5		scmatscmide.0	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6		eqid 2740	. . . . . . . . 9 ⊢ (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅)
7	3, 4, 5, 6	dmatid 22522	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ (𝑁 DMat 𝑅))
8	2, 7	eqeltrid 2848	. . . . . . 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 22530	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
14	10, 13	syldan 590	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
15		simprr 772	. . . . . 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 22530	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
18	16, 17	syldan 590	. . . 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 7461	. . . 4 ⊢ ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 ∗ 1 )(._r‘𝐴)(𝑇 ∗ 1 ))
22	3, 4, 5, 6	dmatmul 22524	. . . 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 2792	. . 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 590	. 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 766	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑁 ∈ Fin)
26		simplr 768	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑅 ∈ Ring)
27	25, 26, 1	3jca 1128	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
28	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
29		3simpc 1150	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
30	3, 11, 5, 2, 12	scmatscmide 22534	. . . . . . 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 583	. . . . . 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 1128	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵))
33	32	3ad2ant1 1133	. . . . . . 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 22534	. . . . . . 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 583	. . . . . 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 7466	. . . . 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 4567	. . . 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 7527	. . 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 4554	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆)
40		iftrue 4554	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇)
41	39, 40	oveq12d 7466	. . . . . . 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 4579	. . . . 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 7527	. . . 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 2741	. . . . . . . 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 2757	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 = 𝑗 ↔ 𝑥 = 𝑦))
47		scmatscmiddistr.t	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑅)
48	47	eqcomi 2749	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = ·
49	48	oveqi 7461	. . . . . . . . . . 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 4572	. . . . . . . . 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 770	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
54		simprr 772	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
55		ovex 7481	. . . . . . . . . 10 ⊢ (𝑆 · 𝑇) ∈ V
56	5	fvexi 6934	. . . . . . . . . 10 ⊢ 0 ∈ V
57	55, 56	ifex 4598	. . . . . . . . 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 7602	. . . . . . 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 1128	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵))
61	11, 47	ringcl 20277	. . . . . . . . . 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 1128	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵))
64	3, 11, 5, 2, 12	scmatscmide 22534	. . . . . . . 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 579	. . . . . . 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 2783	. . . . . 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 3208	. . . . 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 2740	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
69	11, 68	ringcl 20277	. . . . . . . . . 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 20290	. . . . . . . . . . 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 4594	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) ∈ 𝐵)
75	74	3ad2ant1 1133	. . . . . . 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 22450	. . . . . 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 22470	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
78	4, 2	ringidcl 20289	. . . . . . . . . 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 22458	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
83	81, 82	syldan 590	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
84	3, 4	eqmat 22451	. . . . . 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 583	. . . . 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 2780	. . 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 2780	. 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 2780	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 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ifcif 4548 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 Fincfn 9003 Basecbs 17258 ._rcmulr 17312 ·_𝑠 cvsca 17315 0_gc0g 17499 1_rcur 20208 Ringcrg 20260 Mat cmat 22432 DMat cdmat 22515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-subrg 20597 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-mamu 22416 df-mat 22433 df-dmat 22517

This theorem is referenced by: scmatmulcl 22545 scmatmhm 22561

W3C validator