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Theorem assalmod 21755
Description: An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assalmod (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)

Proof of Theorem assalmod
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2726 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2726 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
4 eqid 2726 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
5 eqid 2726 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
61, 2, 3, 4, 5isassa 21751 . . 3 (π‘Š ∈ AssAlg ↔ ((π‘Š ∈ LMod ∧ π‘Š ∈ Ring) ∧ 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 497 . 2 (π‘Š ∈ AssAlg β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring))
87simpld 494 1 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  .rcmulr 17207  Scalarcsca 17209   ·𝑠 cvsca 17210  Ringcrg 20138  LModclmod 20706  AssAlgcasa 21745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-assa 21748
This theorem is referenced by:  assasca  21757  assa2ass  21758  issubassa3  21760  issubassa  21761  assapropd  21766  aspval  21767  asplss  21768  ascldimul  21782  asclrhm  21784  rnascl  21785  issubassa2  21786  aspval2  21792  assamulgscmlem1  21793  assamulgscmlem2  21794  asclmulg  21796  mplmon2mul  21972  mplind  21973  matinv  22534  assaascl0  47336  assaascl1  47337
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