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Theorem scmatmat 21046
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 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 scmatmat.a . . 3 𝐴 = (𝑁 Mat 𝑅)
3 scmatmat.b . . 3 𝐵 = (Base‘𝐴)
4 eqid 2818 . . 3 (1r𝐴) = (1r𝐴)
5 eqid 2818 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
6 scmatmat.s . . 3 𝑆 = (𝑁 ScMat 𝑅)
71, 2, 3, 4, 5, 6scmatel 21042 . 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 254 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  cfv 6348  (class class class)co 7145  Fincfn 8497  Basecbs 16471   ·𝑠 cvsca 16557  1rcur 19180   Mat cmat 20944   ScMat cscmat 21026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-scmat 21028
This theorem is referenced by:  scmatsgrp  21056  scmatcrng  21058
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