Theorem slmdcmn 29758

Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)

Assertion

Ref	Expression
slmdcmn	|- ( W e. SLMod -> W e. 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 2622	. . 3 |- ( Base ` W ) = ( Base ` W )
2		eqid 2622	. . 3 |- ( +g ` W ) = ( +g ` W )
3		eqid 2622	. . 3 |- ( .s ` W ) = ( .s ` W )
4		eqid 2622	. . 3 |- ( 0g ` W ) = ( 0g ` W )
5		eqid 2622	. . 3 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2622	. . 3 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
7		eqid 2622	. . 3 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
8		eqid 2622	. . 3 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
9		eqid 2622	. . 3 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
10		eqid 2622	. . 3 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 29755	. 2 |- ( W e. SLMod <-> ( W e. CMnd /\ (Scalar ` W ) e. SRing /\ A. w e. ( Base ` (Scalar ` W ) ) A. z e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( z ( .s ` W ) y ) e. ( 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 1076	1 |- ( W e. SLMod -> W e. CMnd )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100  CMndccmn 18193   1rcur 18501  SRingcsrg 18505  SLModcslmd 29753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-slmd 29754

This theorem is referenced by:  slmdmnd  29759  gsumvsca1  29782  gsumvsca2  29783