Theorem assasca 19321

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Base ` W ) = ( Base ` W )
2		assasca.f	. . . 4 |- F = (Scalar ` W )
3		eqid 2622	. . . 4 |- ( Base ` F ) = ( Base ` F )
4		eqid 2622	. . . 4 |- ( .s ` W ) = ( .s ` W )
5		eqid 2622	. . . 4 |- ( .r ` W ) = ( .r ` W )
6	1, 2, 3, 4, 5	isassa 19315	. . 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 ) ) ) ) )
7	6	simplbi 476	. 2 |- ( W e. AssAlg -> ( W e. LMod /\ W e. Ring /\ F e. CRing ) )
8	7	simp3d 1075	1 |- ( W e. AssAlg -> F e. CRing )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-assa 19312

This theorem is referenced by:  assa2ass  19322  issubassa  19324  asclrhm  19342  assamulgscmlem2  19349