Theorem slmdcmn 30837

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.

Step	Hyp	Ref	Expression
1		eqid 2824	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2824	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		eqid 2824	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2824	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
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))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 30834	. 2 ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ (Scalar‘𝑊) ∈ SRing ∧ ∀𝑤 ∈ (Base‘(Scalar‘𝑊))∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝑧( ·𝑠 ‘𝑊)(𝑦(+g‘𝑊)𝑥)) = ((𝑧( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑥)) ∧ ((𝑤(+g‘(Scalar‘𝑊))𝑧)( ·𝑠 ‘𝑊)𝑦) = ((𝑤( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦))) ∧ (((𝑤(.r‘(Scalar‘𝑊))𝑧)( ·𝑠 ‘𝑊)𝑦) = (𝑤( ·𝑠 ‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦)) ∧ ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑦) = 𝑦 ∧ ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑦) = (0g‘𝑊)))))
12	11	simp1bi 1141	1 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  +_gcplusg 16568  ._rcmulr 16569  Scalarcsca 16571   ·_𝑠 cvsca 16572  0_gc0g 16716  CMndccmn 18909  1_rcur 19254  SRingcsrg 19258  SLModcslmd 30832

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-nul 5213

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-ov 7162  df-slmd 30833

This theorem is referenced by:  slmdmnd  30838  gsumvsca1  30858  gsumvsca2  30859