MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  assalmod Structured version   Visualization version   GIF version

Theorem assalmod 21415
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 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 21411 . . 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 499 . 2 (π‘Š ∈ AssAlg β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring))
87simpld 496 1 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  Ringcrg 20056  LModclmod 20471  AssAlgcasa 21405
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 5307
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 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-assa 21408
This theorem is referenced by:  assasca  21417  assa2ass  21418  issubassa3  21420  issubassa  21421  assapropd  21426  aspval  21427  asplss  21428  ascldimul  21442  asclrhm  21444  rnascl  21445  issubassa2  21446  aspval2  21452  assamulgscmlem1  21453  assamulgscmlem2  21454  mplmon2mul  21630  mplind  21631  matinv  22179  asclmulg  32635  assaascl0  47060  assaascl1  47061
  Copyright terms: Public domain W3C validator