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Theorem scmatmat 21858
Description: An 𝑁 x 𝑁 scalar matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
scmatmat.a 𝐴 = (𝑁 Mat 𝑅)
scmatmat.b 𝐵 = (Base‘𝐴)
scmatmat.s 𝑆 = (𝑁 ScMat 𝑅)
Assertion
Ref Expression
scmatmat ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))

Proof of Theorem scmatmat
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 scmatmat.a . . 3 𝐴 = (𝑁 Mat 𝑅)
3 scmatmat.b . . 3 𝐵 = (Base‘𝐴)
4 eqid 2736 . . 3 (1r𝐴) = (1r𝐴)
5 eqid 2736 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
6 scmatmat.s . . 3 𝑆 = (𝑁 ScMat 𝑅)
71, 2, 3, 4, 5, 6scmatel 21854 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠𝐴)(1r𝐴)))))
8 simpl 483 . 2 ((𝑀𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠𝐴)(1r𝐴))) → 𝑀𝐵)
97, 8syl6bi 252 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3073  cfv 6496  (class class class)co 7357  Fincfn 8883  Basecbs 17083   ·𝑠 cvsca 17137  1rcur 19913   Mat cmat 21754   ScMat cscmat 21838
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-scmat 21840
This theorem is referenced by:  scmatsgrp  21868  scmatcrng  21870
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