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Theorem slmdcmn 33211
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 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2737 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2737 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2737 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2737 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2737 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7 eqid 2737 . . 3 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
8 eqid 2737 . . 3 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
9 eqid 2737 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
10 eqid 2737 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 33208 . 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 1146 1 (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  CMndccmn 19798  1rcur 20178  SRingcsrg 20183  SLModcslmd 33206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-slmd 33207
This theorem is referenced by:  slmdmnd  33212  gsumvsca1  33232  gsumvsca2  33233
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