Theorem assaring 21415

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 2732	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2732	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2732	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4		eqid 2732	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
5		eqid 2732	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
6	1, 2, 3, 4, 5	isassa 21410	. . 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 498	. 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
8	7	simprd 496	1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  Ringcrg 20055  LModclmod 20470  AssAlgcasa 21404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-assa 21407

This theorem is referenced by:  issubassa  21420  assapropd  21425  aspval  21426  asclmul1  21439  asclmul2  21440  ascldimul  21441  asclrhm  21443  rnascl  21444  aspval2  21451  assamulgscmlem1  21452  assamulgscmlem2  21453  zlmassa  21455  mplind  21630  evlseu  21645  pf1subrg  21866  matinv  22178  asclmulg  32630  irngnzply1lem  32749  assaascl0  47050  assaascl1  47051