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Theorem slmdcmn 30088
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 2760 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2760 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2760 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2760 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2760 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2760 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7 eqid 2760 . . 3 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
8 eqid 2760 . . 3 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
9 eqid 2760 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
10 eqid 2760 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 30085 . 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 1140 1 (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  Scalarcsca 16166   ·𝑠 cvsca 16167  0gc0g 16322  CMndccmn 18413  1rcur 18721  SRingcsrg 18725  SLModcslmd 30083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-slmd 30084
This theorem is referenced by:  slmdmnd  30089  gsumvsca1  30112  gsumvsca2  30113
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