MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  assasca Structured version   Unicode version

Theorem assasca 17840
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
assasca.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
assasca  |-  ( W  e. AssAlg  ->  F  e.  CRing )

Proof of Theorem assasca
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 assasca.f . . . 4  |-  F  =  (Scalar `  W )
3 eqid 2467 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
4 eqid 2467 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2467 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17834 . . 3  |-  ( W  e. AssAlg 
<->  ( ( W  e. 
LMod  /\  W  e.  Ring  /\  F  e.  CRing )  /\  A. z  e.  ( Base `  F ) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( z ( .s `  W ) x ) ( .r `  W
) y )  =  ( z ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( z ( .s `  W ) y ) )  =  ( z ( .s
`  W ) ( x ( .r `  W ) y ) ) ) ) )
76simplbi 460 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  F  e.  CRing ) )
87simp3d 1010 1  |-  ( W  e. AssAlg  ->  F  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   ` cfv 5594  (class class class)co 6295   Basecbs 14507   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   Ringcrg 17070   CRingccrg 17071   LModclmod 17383  AssAlgcasa 17828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-assa 17831
This theorem is referenced by:  assa2ass  17841  issubassa  17843  asclrhm  17861  assamulgscmlem2  17868
  Copyright terms: Public domain W3C validator