Theorem assaring 21843

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 2740	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2740	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2740	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21838	. . 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 496	1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  Scalarcsca 17221   ·_𝑠 cvsca 17222  Ringcrg 20212  LModclmod 20857  AssAlgcasa 21832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-assa 21835

This theorem is referenced by:  issubassa  21849  assapropd  21853  aspval  21854  asclelbas  21865  asclmul1  21868  asclmul2  21869  ascldimul  21870  asclrhm  21872  rnascl  21873  aspval2  21880  assamulgscmlem1  21881  assamulgscmlem2  21882  asclmulg  21884  zlmassa  21885  mplind  22053  evlseu  22066  pf1subrg  22341  matinv  22667  lactlmhm  33825  assalactf1o  33826  assarrginv  33827  irngnzply1lem  33881  assaascl0  48879  assaascl1  48880  asclelbasALT  49503