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Theorem assalmod 21421
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 2732 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2732 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2732 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
4 eqid 2732 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
5 eqid 2732 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
61, 2, 3, 4, 5isassa 21417 . . 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 498 . 2 (π‘Š ∈ AssAlg β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring))
87simpld 495 1 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  Ringcrg 20058  LModclmod 20475  AssAlgcasa 21411
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 7414  df-assa 21414
This theorem is referenced by:  assasca  21423  assa2ass  21424  issubassa3  21426  issubassa  21427  assapropd  21432  aspval  21433  asplss  21434  ascldimul  21448  asclrhm  21450  rnascl  21451  issubassa2  21452  aspval2  21458  assamulgscmlem1  21459  assamulgscmlem2  21460  mplmon2mul  21636  mplind  21637  matinv  22186  asclmulg  32680  assaascl0  47139  assaascl1  47140
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