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Theorem slmdcmn 29089
Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
slmdcmn (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)

Proof of Theorem slmdcmn
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2610 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2610 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2610 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2610 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2610 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7 eqid 2610 . . 3 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
8 eqid 2610 . . 3 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
9 eqid 2610 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
10 eqid 2610 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 29086 . 2 (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ (Scalar‘𝑊) ∈ SRing ∧ ∀𝑤 ∈ (Base‘(Scalar‘𝑊))∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝑧( ·𝑠𝑊)(𝑦(+g𝑊)𝑥)) = ((𝑧( ·𝑠𝑊)𝑦)(+g𝑊)(𝑧( ·𝑠𝑊)𝑥)) ∧ ((𝑤(+g‘(Scalar‘𝑊))𝑧)( ·𝑠𝑊)𝑦) = ((𝑤( ·𝑠𝑊)𝑦)(+g𝑊)(𝑧( ·𝑠𝑊)𝑦))) ∧ (((𝑤(.r‘(Scalar‘𝑊))𝑧)( ·𝑠𝑊)𝑦) = (𝑤( ·𝑠𝑊)(𝑧( ·𝑠𝑊)𝑦)) ∧ ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑦) = 𝑦 ∧ ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑦) = (0g𝑊)))))
1211simp1bi 1069 1 (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  CMndccmn 18016  1rcur 18324  SRingcsrg 18328  SLModcslmd 29084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-slmd 29085
This theorem is referenced by:  slmdmnd  29090  gsumvsca1  29113  gsumvsca2  29114
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