Theorem assalmod 21808

Description: An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)

Assertion

Ref	Expression
assalmod	⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)

Proof of Theorem assalmod

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2728	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2728	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2728	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2728	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21804	. . 3 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (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 496	. 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
8	7	simpld 493	1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ‘cfv 6553  (class class class)co 7426  Basecbs 17189  ._rcmulr 17243  Scalarcsca 17245   ·_𝑠 cvsca 17246  Ringcrg 20187  LModclmod 20757  AssAlgcasa 21798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-assa 21801

This theorem is referenced by:  assasca  21810  assa2ass  21811  issubassa3  21813  issubassa  21814  assapropd  21819  aspval  21820  asplss  21821  ascldimul  21835  asclrhm  21837  rnascl  21838  issubassa2  21839  aspval2  21845  assamulgscmlem1  21846  assamulgscmlem2  21847  asclmulg  21849  mplmon2mul  22030  mplind  22031  matinv  22607  assaascl0  47544  assaascl1  47545