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Theorem scmatscmiddistr 22878

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 783	. . . . . 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 2763	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
5		scmatscmide.0	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6		eqid 2763	. . . . . . . . 9 ⊢ (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅)
7	3, 4, 5, 6	dmatid 22865	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ (𝑁 DMat 𝑅))
8	2, 7	eqeltrid 2868	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (𝑁 DMat 𝑅))
9	8	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (𝑁 DMat 𝑅))
10	1, 9	jca 521	. . . . 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 22873	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
14	10, 13	syldan 603	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
15		simprr 785	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑇 ∈ 𝐵)
16	15, 9	jca 521	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅)))
17	11, 3, 4, 12, 6	dmatscmcl 22873	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
18	16, 17	syldan 603	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))
19	14, 18	jca 521	. . 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 7464	. . . 4 ⊢ ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 ∗ 1 )(._r‘𝐴)(𝑇 ∗ 1 ))
22	3, 4, 5, 6	dmatmul 22867	. . . 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 2810	. . 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 603	. 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 779	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑁 ∈ Fin)
26		simplr 781	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑅 ∈ Ring)
27	25, 26, 1	3jca 1146	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
28	27	3ad2ant1 1151	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵))
29		3simpc 1168	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
30	3, 11, 5, 2, 12	scmatscmide 22877	. . . . . . 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 596	. . . . . 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 1146	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵))
33	32	3ad2ant1 1151	. . . . . . 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 22877	. . . . . . 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 596	. . . . . 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 7469	. . . . 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 4505	. . . 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 7528	. . 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 4491	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆)
40		iftrue 4491	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇)
41	39, 40	oveq12d 7469	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (if(𝑖 = 𝑗, 𝑆, 0 )(._r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(._r‘𝑅)𝑇))
42	41	adantl 487	. . . . . 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 4517	. . . . 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 7528	. . . 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 2764	. . . . . . . 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 2780	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 = 𝑗 ↔ 𝑥 = 𝑦))
47		scmatscmiddistr.t	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑅)
48	47	eqcomi 2772	. . . . . . . . . . . 12 ⊢ (._r‘𝑅) = ·
49	48	oveqi 7464	. . . . . . . . . . 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 4510	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
52	51	adantl 487	. . . . . . . 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 783	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
54		simprr 785	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
55		ovex 7484	. . . . . . . . . 10 ⊢ (𝑆 · 𝑇) ∈ V
56	5	fvexi 6930	. . . . . . . . . 10 ⊢ 0 ∈ V
57	55, 56	ifex 4536	. . . . . . . . 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 7604	. . . . . . 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 1146	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵))
61	11, 47	ringcl 20528	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆 · 𝑇) ∈ 𝐵)
62	60, 61	syl 18	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 · 𝑇) ∈ 𝐵)
63	25, 26, 62	3jca 1146	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵))
64	3, 11, 5, 2, 12	scmatscmide 22877	. . . . . . . 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 592	. . . . . . 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 2801	. . . . . 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 3209	. . . . 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 2763	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
69	11, 68	ringcl 20528	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆(._r‘𝑅)𝑇) ∈ 𝐵)
70	60, 69	syl 18	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆(._r‘𝑅)𝑇) ∈ 𝐵)
71	11, 5	ring0cl 20547	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
72	71	adantl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐵)
73	72	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 0 ∈ 𝐵)
74	70, 73	ifcld 4532	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → if(𝑖 = 𝑗, (𝑆(._r‘𝑅)𝑇), 0 ) ∈ 𝐵)
75	74	3ad2ant1 1151	. . . . . . 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 22793	. . . . . 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 22813	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
78	4, 2	ringidcl 20545	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
79	77, 78	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
80	79	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (Base‘𝐴))
81	62, 80	jca 521	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴)))
82	11, 3, 4, 12	matvscl 22801	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
83	81, 82	syldan 603	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴))
84	3, 4	eqmat 22794	. . . . . 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 596	. . . . 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 260	. . . 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 2798	. . 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 2798	. 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 2798	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ifcif 4485 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 Fincfn 9011 Basecbs 17431 ._rcmulr 17473 ·_𝑠 cvsca 17476 0_gc0g 17660 1_rcur 20458 Ringcrg 20510 Mat cmat 22777 DMat cdmat 22858

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-ot 4864 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-fz 13680 df-fzo 13829 df-seq 14185 df-hash 14516 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-hom 17497 df-cco 17498 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-subrg 20873 df-lmod 21188 df-lss 21258 df-sra 21499 df-rgmod 21500 df-dsmm 22093 df-frlm 22108 df-mamu 22761 df-mat 22778 df-dmat 22860

This theorem is used by: scmatmulcl 22888 scmatmhm 22904

W3C validator