Theorem assaring 21802

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 2733	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2733	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2733	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2733	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21797	. . 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 2113  ∀wral 3048  ‘cfv 6488  (class class class)co 7354  Basecbs 17124  ._rcmulr 17166  Scalarcsca 17168   ·_𝑠 cvsca 17169  Ringcrg 20155  LModclmod 20797  AssAlgcasa 21791

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 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

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 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-assa 21794

This theorem is referenced by:  issubassa  21808  assapropd  21813  aspval  21814  asclmul1  21827  asclmul2  21828  ascldimul  21829  asclrhm  21831  rnascl  21832  aspval2  21839  assamulgscmlem1  21840  assamulgscmlem2  21841  asclmulg  21843  zlmassa  21844  mplind  22008  evlseu  22021  pf1subrg  22266  matinv  22595  lactlmhm  33670  assalactf1o  33671  assarrginv  33672  irngnzply1lem  33726  assaascl0  48508  assaascl1  48509  asclelbas  49133  asclelbasALT  49134