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Theorem assaring 21283
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assaring (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)

Proof of Theorem assaring
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2733 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
4 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
5 eqid 2733 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
61, 2, 3, 4, 5isassa 21278 . . 3 (π‘Š ∈ AssAlg ↔ ((π‘Š ∈ LMod ∧ π‘Š ∈ Ring ∧ (Scalarβ€˜π‘Š) ∈ CRing) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((𝑧( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (𝑧( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦)) = (𝑧( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
76simplbi 499 . 2 (π‘Š ∈ AssAlg β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring ∧ (Scalarβ€˜π‘Š) ∈ CRing))
87simp2d 1144 1 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Ringcrg 19969  CRingccrg 19970  LModclmod 20336  AssAlgcasa 21272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-assa 21275
This theorem is referenced by:  issubassa  21288  assapropd  21291  aspval  21292  asclmul1  21305  asclmul2  21306  ascldimul  21307  asclrhm  21309  rnascl  21310  aspval2  21317  assamulgscmlem1  21318  assamulgscmlem2  21319  zlmassa  21321  mplind  21494  evlseu  21509  pf1subrg  21730  matinv  22042  asclmulg  32311  irngnzply1lem  32421  assaascl0  46546  assaascl1  46547
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