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Theorem assasca 16383
 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 2438 . . . 4
2 assasca.f . . . 4 Scalar
3 eqid 2438 . . . 4
4 eqid 2438 . . . 4
5 eqid 2438 . . . 4
61, 2, 3, 4, 5isassa 16377 . . 3 AssAlg
76simplbi 448 . 2 AssAlg
87simp3d 972 1 AssAlg
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cfv 5456  (class class class)co 6083  cbs 13471  cmulr 13532  Scalarcsca 13534  cvsca 13535  crg 15662  ccrg 15663  clmod 15952  AssAlgcasa 16371 This theorem is referenced by:  issubassa  16385  asclrhm  16402 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-assa 16374
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