Theorem assalmod 22040

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 2766	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2766	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2766	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2766	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2766	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 22036	. . 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 502	. 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
8	7	simpld 500	1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  ._rcmulr 17336  Scalarcsca 17338   ·_𝑠 cvsca 17339  Ringcrg 20340  LModclmod 21011  AssAlgcasa 22030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-assa 22033

This theorem is used by:  assasca  22042  assa2ass  22043  assa2ass2  22044  issubassa3  22046  issubassa  22047  assapropd  22051  aspval  22052  asplss  22053  asclelbas  22063  ascldimul  22068  asclrhm  22070  rnascl  22071  issubassa2  22072  aspval2  22078  assamulgscmlem1  22079  assamulgscmlem2  22080  asclmulg  22082  mplmon2mul  22250  mplind  22251  matinv  22864  lactlmhm  34048  assalactf1o  34049  assaascl0  49194  assaascl1  49195  asclelbasALT  49817