Theorem lmodslmd 29757

Description: Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)

Assertion

Ref	Expression
lmodslmd	|- ( W e. LMod -> W e. SLMod )

Proof of Theorem lmodslmd

Dummy variables r q w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmodcmn 18911	. 2 |- ( W e. LMod -> W e. CMnd )
2		eqid 2622	. . . 4 |- (Scalar ` W ) = (Scalar ` W )
3	2	lmodring 18871	. . 3 |- ( W e. LMod -> (Scalar ` W ) e. Ring )
4		ringsrg 18589	. . 3 |- ( (Scalar ` W ) e. Ring -> (Scalar ` W ) e. SRing )
5	3, 4	syl 17	. 2 |- ( W e. LMod -> (Scalar ` W ) e. SRing )
6		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` W ) = ( Base ` W )
7		eqid 2622	. . . . . . . . . . . . . 14 |- ( +g ` W ) = ( +g ` W )
8		eqid 2622	. . . . . . . . . . . . . 14 |- ( .s ` W ) = ( .s ` W )
9		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
10		eqid 2622	. . . . . . . . . . . . . 14 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
11		eqid 2622	. . . . . . . . . . . . . 14 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
12		eqid 2622	. . . . . . . . . . . . . 14 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
13	6, 7, 8, 2, 9, 10, 11, 12	islmod 18867	. . . . . . . . . . . . 13 |- ( W e. LMod <-> ( W e. Grp /\ (Scalar ` W ) e. Ring /\ A. q e. ( Base ` (Scalar ` W ) ) A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) ) )
14	13	simp3bi 1078	. . . . . . . . . . . 12 |- ( W e. LMod -> A. q e. ( Base ` (Scalar ` W ) ) A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) )
15	14	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) -> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) )
16	15	r19.21bi 2932	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) -> A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) )
17	16	r19.21bi 2932	. . . . . . . . 9 |- ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) -> A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) )
18	17	r19.21bi 2932	. . . . . . . 8 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) ) )
19	18	simpld 475	. . . . . . 7 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) )
20	18	simprd 479	. . . . . . . . 9 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w ) )
21	20	simpld 475	. . . . . . . 8 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) )
22	20	simprd 479	. . . . . . . 8 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w )
23		simp-4l 806	. . . . . . . . 9 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> W e. LMod )
24		eqid 2622	. . . . . . . . . 10 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
25		eqid 2622	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
26	6, 2, 8, 24, 25	lmod0vs 18896	. . . . . . . . 9 |- ( ( W e. LMod /\ w e. ( Base ` W ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) )
27	23, 26	sylancom 701	. . . . . . . 8 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) )
28	21, 22, 27	3jca 1242	. . . . . . 7 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) )
29	19, 28	jca 554	. . . . . 6 |- ( ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) /\ w e. ( Base ` W ) ) -> ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) )
30	29	ralrimiva 2966	. . . . 5 |- ( ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ x e. ( Base ` W ) ) -> A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) )
31	30	ralrimiva 2966	. . . 4 |- ( ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) /\ r e. ( Base ` (Scalar ` W ) ) ) -> A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) )
32	31	ralrimiva 2966	. . 3 |- ( ( W e. LMod /\ q e. ( Base ` (Scalar ` W ) ) ) -> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) )
33	32	ralrimiva 2966	. 2 |- ( W e. LMod -> A. q e. ( Base ` (Scalar ` W ) ) A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) )
34	6, 7, 8, 25, 2, 9, 10, 11, 12, 24	isslmd 29755	. 2 |- ( W e. SLMod <-> ( W e. CMnd /\ (Scalar ` W ) e. SRing /\ A. q e. ( Base ` (Scalar ` W ) ) A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. w e. ( Base ` W ) ( ( ( r ( .s ` W ) w ) e. ( Base ` W ) /\ ( r ( .s ` W ) ( w ( +g ` W ) x ) ) = ( ( r ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) x ) ) /\ ( ( q ( +g ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( ( q ( .s ` W ) w ) ( +g ` W ) ( r ( .s ` W ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` W ) ) r ) ( .s ` W ) w ) = ( q ( .s ` W ) ( r ( .s ` W ) w ) ) /\ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) w ) = w /\ ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) w ) = ( 0g ` W ) ) ) ) )
35	1, 5, 33, 34	syl3anbrc 1246	1 |- ( W e. LMod -> W e. SLMod )

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   Grpcgrp 17422  CMndccmn 18193   1rcur 18501  SRingcsrg 18505   Ringcrg 18547   LModclmod 18863  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-lmod 18865  df-slmd 29754

This theorem is referenced by: (None)