Theorem assaring 21899

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

Assertion

Ref	Expression
assaring	⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Proof of Theorem assaring

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

Step	Hyp	Ref	Expression
1		eqid 2735	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2735	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2735	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2735	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2735	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21894	. . 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 497	. 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
8	7	simprd 495	1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  Ringcrg 20251  LModclmod 20875  AssAlgcasa 21888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-assa 21891

This theorem is referenced by:  issubassa  21905  assapropd  21910  aspval  21911  asclmul1  21924  asclmul2  21925  ascldimul  21926  asclrhm  21928  rnascl  21929  aspval2  21936  assamulgscmlem1  21937  assamulgscmlem2  21938  asclmulg  21940  zlmassa  21941  mplind  22112  evlseu  22125  pf1subrg  22368  matinv  22699  lactlmhm  33662  assalactf1o  33663  assarrginv  33664  irngnzply1lem  33705  assaascl0  48226  assaascl1  48227  asclelbas  48795  asclelbasALT  48796