Theorem assaring 21799

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 2731	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2731	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2731	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2731	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21794	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165  Ringcrg 20152  LModclmod 20794  AssAlgcasa 21788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-assa 21791

This theorem is referenced by:  issubassa  21805  assapropd  21810  aspval  21811  asclmul1  21824  asclmul2  21825  ascldimul  21826  asclrhm  21828  rnascl  21829  aspval2  21836  assamulgscmlem1  21837  assamulgscmlem2  21838  asclmulg  21840  zlmassa  21841  mplind  22006  evlseu  22019  pf1subrg  22264  matinv  22593  lactlmhm  33645  assalactf1o  33646  assarrginv  33647  irngnzply1lem  33701  assaascl0  48418  assaascl1  48419  asclelbas  49043  asclelbasALT  49044