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Theorem scmatmat 20134
 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 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 scmatmat.a . . 3 𝐴 = (𝑁 Mat 𝑅)
3 scmatmat.b . . 3 𝐵 = (Base‘𝐴)
4 eqid 2610 . . 3 (1r𝐴) = (1r𝐴)
5 eqid 2610 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
6 scmatmat.s . . 3 𝑆 = (𝑁 ScMat 𝑅)
71, 2, 3, 4, 5, 6scmatel 20130 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠𝐴)(1r𝐴)))))
8 simpl 472 . 2 ((𝑀𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠𝐴)(1r𝐴))) → 𝑀𝐵)
97, 8syl6bi 242 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695   ·𝑠 cvsca 15772  1rcur 18324   Mat cmat 20032   ScMat cscmat 20114 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-scmat 20116 This theorem is referenced by:  scmatsgrp  20144  scmatcrng  20146
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