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Theorem lmodslmd 27620
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  q 
r  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodcmn 17432 . 2  |-  ( W  e.  LMod  ->  W  e. CMnd
)
2 eqid 2443 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
32lmodring 17394 . . 3  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
4 ringsrg 17111 . . 3  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e. SRing )
53, 4syl 16 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  e. SRing )
6 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  W )  =  (
Base `  W )
7 eqid 2443 . . . . . . . . . . . . . 14  |-  ( +g  `  W )  =  ( +g  `  W )
8 eqid 2443 . . . . . . . . . . . . . 14  |-  ( .s
`  W )  =  ( .s `  W
)
9 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
10 eqid 2443 . . . . . . . . . . . . . 14  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
11 eqid 2443 . . . . . . . . . . . . . 14  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
12 eqid 2443 . . . . . . . . . . . . . 14  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
136, 7, 8, 2, 9, 10, 11, 12islmod 17390 . . . . . . . . . . . . 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 ) ) ) )
1413simp3bi 1014 . . . . . . . . . . . 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 ) ) )
1514r19.21bi 2812 . . . . . . . . . . 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 ) ) )
1615r19.21bi 2812 . . . . . . . . . 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 ) ) )
1716r19.21bi 2812 . . . . . . . . 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 ) ) )
1817r19.21bi 2812 . . . . . . . 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 ) ) )
1918simpld 459 . . . . . . 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 ) ) ) )
2018simprd 463 . . . . . . . . 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 ) )
2120simpld 459 . . . . . . . 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 ) ) )
2220simprd 463 . . . . . . . 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 767 . . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
266, 2, 8, 24, 25lmod0vs 17419 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  w  e.  ( Base `  W
) )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) w )  =  ( 0g `  W ) )
2723, 26sylancom 667 . . . . . . . 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
) )
2821, 22, 273jca 1177 . . . . . . 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 ) ) )
2919, 28jca 532 . . . . . 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 ) ) ) )
3029ralrimiva 2857 . . . . 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 ) ) ) )
3130ralrimiva 2857 . . . 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 ) ) ) )
3231ralrimiva 2857 . . 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 ) ) ) )
3332ralrimiva 2857 . 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 ) ) ) )
346, 7, 8, 25, 2, 9, 10, 11, 12, 24isslmd 27618 . 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 ) ) ) ) )
351, 5, 33, 34syl3anbrc 1181 1  |-  ( W  e.  LMod  ->  W  e. SLMod
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   .rcmulr 14575  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   Grpcgrp 15927  CMndccmn 16672   1rcur 17027  SRingcsrg 17031   Ringcrg 17072   LModclmod 17386  SLModcslmd 27616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-srg 17032  df-ring 17074  df-lmod 17388  df-slmd 27617
This theorem is referenced by: (None)
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