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Theorem assasca 17838
 Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
assasca.f Scalar
Assertion
Ref Expression
assasca AssAlg

Proof of Theorem assasca
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . . . 4
2 assasca.f . . . 4 Scalar
3 eqid 2441 . . . 4
4 eqid 2441 . . . 4
5 eqid 2441 . . . 4
61, 2, 3, 4, 5isassa 17832 . . 3 AssAlg
76simplbi 460 . 2 AssAlg
87simp3d 1009 1 AssAlg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 972   wceq 1381   wcel 1802  wral 2791  cfv 5574  (class class class)co 6277  cbs 14504  cmulr 14570  Scalarcsca 14572  cvsca 14573  crg 17066  ccrg 17067  clmod 17380  AssAlgcasa 17826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280  df-assa 17829 This theorem is referenced by:  assa2ass  17839  issubassa  17841  asclrhm  17859  assamulgscmlem2  17866
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