Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  slmdcmn Structured version   Visualization version   GIF version

Theorem slmdcmn 30865
 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 2824 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2824 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2824 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2824 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2824 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
6 eqid 2824 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7 eqid 2824 . . 3 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
8 eqid 2824 . . 3 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
9 eqid 2824 . . 3 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
10 eqid 2824 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 30862 . 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 1142 1 (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  CMndccmn 18906  1rcur 19251  SRingcsrg 19255  SLModcslmd 30860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-slmd 30861 This theorem is referenced by:  slmdmnd  30866  gsumvsca1  30886  gsumvsca2  30887
 Copyright terms: Public domain W3C validator