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Theorem slmdcmn 32328
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 2733 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2733 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2733 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2733 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2733 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2733 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7 eqid 2733 . . 3 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
8 eqid 2733 . . 3 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
9 eqid 2733 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
10 eqid 2733 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 32325 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cfv 6540  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  CMndccmn 19641  1rcur 19996  SRingcsrg 20000  SLModcslmd 32323
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 5305
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 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407  df-slmd 32324
This theorem is referenced by:  slmdmnd  32329  gsumvsca1  32349  gsumvsca2  32350
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