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Theorem scmatel 19297
 Description: An x scalar matrix over (a ring) . (Contributed by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
scmatval.k
scmatval.a Mat
scmatval.b
scmatval.1
scmatval.t
scmatval.s ScMat
Assertion
Ref Expression
scmatel
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()

Proof of Theorem scmatel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 scmatval.k . . . 4
2 scmatval.a . . . 4 Mat
3 scmatval.b . . . 4
4 scmatval.1 . . . 4
5 scmatval.t . . . 4
6 scmatval.s . . . 4 ScMat
71, 2, 3, 4, 5, 6scmatval 19296 . . 3
87eleq2d 2472 . 2
9 eqeq1 2406 . . . 4
109rexbidv 2917 . . 3
1110elrab 3206 . 2
128, 11syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  wrex 2754  crab 2757  cfv 5568  (class class class)co 6277  cfn 7553  cbs 14839  cvsca 14911  cur 17471   Mat cmat 19199   ScMat cscmat 19281 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-scmat 19283 This theorem is referenced by:  scmatscmid  19298  scmatmat  19301  scmatid  19306  scmataddcl  19308  scmatsubcl  19309  scmatmulcl  19310  smatvscl  19316  scmatrhmcl  19320  mat0scmat  19330  mat1scmat  19331  chmaidscmat  19639
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