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Theorem scmatscmiddistr 20133
 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 = (0g𝑅)
scmatscmide.1 1 = (1r𝐴)
scmatscmide.m = ( ·𝑠𝐴)
scmatscmiddistr.t · = (.r𝑅)
scmatscmiddistr.m × = (.r𝐴)
Assertion
Ref Expression
scmatscmiddistr (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((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.
StepHypRef Expression
1 simprl 790 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 𝑆𝐵)
2 scmatscmide.1 . . . . . . . 8 1 = (1r𝐴)
3 scmatscmide.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
4 eqid 2610 . . . . . . . . 9 (Base‘𝐴) = (Base‘𝐴)
5 scmatscmide.0 . . . . . . . . 9 0 = (0g𝑅)
6 eqid 2610 . . . . . . . . 9 (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅)
73, 4, 5, 6dmatid 20120 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) ∈ (𝑁 DMat 𝑅))
82, 7syl5eqel 2692 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (𝑁 DMat 𝑅))
98adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 1 ∈ (𝑁 DMat 𝑅))
101, 9jca 553 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑆𝐵1 ∈ (𝑁 DMat 𝑅)))
11 scmatscmide.b . . . . . 6 𝐵 = (Base‘𝑅)
12 scmatscmide.m . . . . . 6 = ( ·𝑠𝐴)
1311, 3, 4, 12, 6dmatscmcl 20128 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵1 ∈ (𝑁 DMat 𝑅))) → (𝑆 1 ) ∈ (𝑁 DMat 𝑅))
1410, 13syldan 486 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑆 1 ) ∈ (𝑁 DMat 𝑅))
15 simprr 792 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 𝑇𝐵)
1615, 9jca 553 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑇𝐵1 ∈ (𝑁 DMat 𝑅)))
1711, 3, 4, 12, 6dmatscmcl 20128 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇𝐵1 ∈ (𝑁 DMat 𝑅))) → (𝑇 1 ) ∈ (𝑁 DMat 𝑅))
1816, 17syldan 486 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑇 1 ) ∈ (𝑁 DMat 𝑅))
1914, 18jca 553 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 1 ) ∈ (𝑁 DMat 𝑅)))
20 scmatscmiddistr.m . . . . 5 × = (.r𝐴)
2120oveqi 6562 . . . 4 ((𝑆 1 ) × (𝑇 1 )) = ((𝑆 1 )(.r𝐴)(𝑇 1 ))
223, 4, 5, 6dmatmul 20122 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 1 )(.r𝐴)(𝑇 1 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 )))
2321, 22syl5eq 2656 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 1 ) × (𝑇 1 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 )))
2419, 23syldan 486 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 1 ) × (𝑇 1 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 )))
25 simpll 786 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 𝑁 ∈ Fin)
26 simplr 788 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 𝑅 ∈ Ring)
2725, 26, 13jca 1235 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆𝐵))
28273ad2ant1 1075 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆𝐵))
29 3simpc 1053 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
303, 11, 5, 2, 12scmatscmide 20132 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑆 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 ))
3128, 29, 30syl2anc 691 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑆 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 ))
3225, 26, 153jca 1235 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇𝐵))
33323ad2ant1 1075 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇𝐵))
343, 11, 5, 2, 12scmatscmide 20132 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 ))
3533, 29, 34syl2anc 691 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑇 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 ))
3631, 35oveq12d 6567 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)) = (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )))
3736ifeq1d 4054 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 ) = if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ))
3837mpt2eq3dva 6617 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )))
39 iftrue 4042 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆)
40 iftrue 4042 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇)
4139, 40oveq12d 6567 . . . . . . 7 (𝑖 = 𝑗 → (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(.r𝑅)𝑇))
4241adantl 481 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑖 = 𝑗) → (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(.r𝑅)𝑇))
4342ifeq1da 4066 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ) = if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))
4443mpt2eq3dva 6617 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )))
45 eqidd 2611 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )))
46 eqeq12 2623 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖 = 𝑗𝑥 = 𝑦))
47 scmatscmiddistr.t . . . . . . . . . . . . 13 · = (.r𝑅)
4847eqcomi 2619 . . . . . . . . . . . 12 (.r𝑅) = ·
4948oveqi 6562 . . . . . . . . . . 11 (𝑆(.r𝑅)𝑇) = (𝑆 · 𝑇)
5049a1i 11 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑆(.r𝑅)𝑇) = (𝑆 · 𝑇))
5146, 50ifbieq1d 4059 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
5251adantl 481 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑖 = 𝑥𝑗 = 𝑦)) → if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
53 simprl 790 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
54 simprr 792 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
55 ovex 6577 . . . . . . . . . 10 (𝑆 · 𝑇) ∈ V
56 fvex 6113 . . . . . . . . . . 11 (0g𝑅) ∈ V
575, 56eqeltri 2684 . . . . . . . . . 10 0 ∈ V
5855, 57ifex 4106 . . . . . . . . 9 if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈ V
5958a1i 11 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈ V)
6045, 52, 53, 54, 59ovmpt2d 6686 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
6126, 1, 153jca 1235 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑅 ∈ Ring ∧ 𝑆𝐵𝑇𝐵))
6211, 47ringcl 18384 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆𝐵𝑇𝐵) → (𝑆 · 𝑇) ∈ 𝐵)
6361, 62syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑆 · 𝑇) ∈ 𝐵)
6425, 26, 633jca 1235 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵))
653, 11, 5, 2, 12scmatscmide 20132 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥((𝑆 · 𝑇) 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
6664, 65sylan 487 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥((𝑆 · 𝑇) 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ))
6760, 66eqtr4d 2647 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) 1 )𝑦))
6867ralrimivva 2954 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) 1 )𝑦))
69 eqid 2610 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
7011, 69ringcl 18384 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆𝐵𝑇𝐵) → (𝑆(.r𝑅)𝑇) ∈ 𝐵)
7161, 70syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑆(.r𝑅)𝑇) ∈ 𝐵)
7211, 5ring0cl 18392 . . . . . . . . . . 11 (𝑅 ∈ Ring → 0𝐵)
7372adantl 481 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐵)
7473adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 0𝐵)
7571, 74ifcld 4081 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ) ∈ 𝐵)
76753ad2ant1 1075 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ) ∈ 𝐵)
773, 11, 4, 25, 26, 76matbas2d 20048 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) ∈ (Base‘𝐴))
783matring 20068 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
794, 2ringidcl 18391 . . . . . . . . . 10 (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
8078, 79syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
8180adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → 1 ∈ (Base‘𝐴))
8263, 81jca 553 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 · 𝑇) ∈ 𝐵1 ∈ (Base‘𝐴)))
8311, 3, 4, 12matvscl 20056 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) 1 ) ∈ (Base‘𝐴))
8482, 83syldan 486 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 · 𝑇) 1 ) ∈ (Base‘𝐴))
853, 4eqmat 20049 . . . . . 6 (((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) ∈ (Base‘𝐴) ∧ ((𝑆 · 𝑇) 1 ) ∈ (Base‘𝐴)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) 1 ) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) 1 )𝑦)))
8677, 84, 85syl2anc 691 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) 1 ) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) 1 )𝑦)))
8768, 86mpbird 246 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) 1 ))
8844, 87eqtrd 2644 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0 )(.r𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = ((𝑆 · 𝑇) 1 ))
8938, 88eqtrd 2644 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 1 )𝑗)(.r𝑅)(𝑖(𝑇 1 )𝑗)), 0 )) = ((𝑆 · 𝑇) 1 ))
9024, 89eqtrd 2644 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 1 ) × (𝑇 1 )) = ((𝑆 · 𝑇) 1 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ifcif 4036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  Basecbs 15695  .rcmulr 15769   ·𝑠 cvsca 15772  0gc0g 15923  1rcur 18324  Ringcrg 18370   Mat cmat 20032   DMat cdmat 20113 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-dmat 20115 This theorem is referenced by:  scmatmulcl  20143  scmatmhm  20159
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