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Theorem lmodvsinv2 15810
Description: Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
lmodvsinv2.b  |-  B  =  ( Base `  W
)
lmodvsinv2.f  |-  F  =  (Scalar `  W )
lmodvsinv2.s  |-  .x.  =  ( .s `  W )
lmodvsinv2.n  |-  N  =  ( inv g `  W )
lmodvsinv2.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
lmodvsinv2  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( N `  X ) )  =  ( N `  ( R  .x.  X ) ) )

Proof of Theorem lmodvsinv2
StepHypRef Expression
1 simp1 955 . . . . . . 7  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  W  e.  LMod )
2 lmodgrp 15650 . . . . . . 7  |-  ( W  e.  LMod  ->  W  e. 
Grp )
31, 2syl 15 . . . . . 6  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  W  e.  Grp )
4 simp3 957 . . . . . 6  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  X  e.  B )
5 lmodvsinv2.b . . . . . . 7  |-  B  =  ( Base `  W
)
6 eqid 2296 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
7 eqid 2296 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
8 lmodvsinv2.n . . . . . . 7  |-  N  =  ( inv g `  W )
95, 6, 7, 8grprinv 14545 . . . . . 6  |-  ( ( W  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  W ) ( N `
 X ) )  =  ( 0g `  W ) )
103, 4, 9syl2anc 642 . . . . 5  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
1110oveq2d 5890 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( X ( +g  `  W ) ( N `  X
) ) )  =  ( R  .x.  ( 0g `  W ) ) )
12 simp2 956 . . . . 5  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  R  e.  K )
135, 8grpinvcl 14543 . . . . . 6  |-  ( ( W  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
143, 4, 13syl2anc 642 . . . . 5  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( N `  X )  e.  B )
15 lmodvsinv2.f . . . . . 6  |-  F  =  (Scalar `  W )
16 lmodvsinv2.s . . . . . 6  |-  .x.  =  ( .s `  W )
17 lmodvsinv2.k . . . . . 6  |-  K  =  ( Base `  F
)
185, 6, 15, 16, 17lmodvsdi 15666 . . . . 5  |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  X  e.  B  /\  ( N `  X )  e.  B ) )  ->  ( R  .x.  ( X ( +g  `  W
) ( N `  X ) ) )  =  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  ( N `  X )
) ) )
191, 12, 4, 14, 18syl13anc 1184 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( X ( +g  `  W ) ( N `  X
) ) )  =  ( ( R  .x.  X ) ( +g  `  W ) ( R 
.x.  ( N `  X ) ) ) )
2015, 16, 17, 7lmodvs0 15680 . . . . 5  |-  ( ( W  e.  LMod  /\  R  e.  K )  ->  ( R  .x.  ( 0g `  W ) )  =  ( 0g `  W
) )
211, 12, 20syl2anc 642 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( 0g `  W ) )  =  ( 0g `  W
) )
2211, 19, 213eqtr3d 2336 . . 3  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  (
( R  .x.  X
) ( +g  `  W
) ( R  .x.  ( N `  X ) ) )  =  ( 0g `  W ) )
235, 15, 16, 17lmodvscl 15660 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  X )  e.  B )
245, 15, 16, 17lmodvscl 15660 . . . . 5  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  ( N `  X )  e.  B )  ->  ( R  .x.  ( N `  X ) )  e.  B )
251, 12, 14, 24syl3anc 1182 . . . 4  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( N `  X ) )  e.  B )
265, 6, 7, 8grpinvid1 14546 . . . 4  |-  ( ( W  e.  Grp  /\  ( R  .x.  X )  e.  B  /\  ( R  .x.  ( N `  X ) )  e.  B )  ->  (
( N `  ( R  .x.  X ) )  =  ( R  .x.  ( N `  X ) )  <->  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  ( N `  X )
) )  =  ( 0g `  W ) ) )
273, 23, 25, 26syl3anc 1182 . . 3  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  (
( N `  ( R  .x.  X ) )  =  ( R  .x.  ( N `  X ) )  <->  ( ( R 
.x.  X ) ( +g  `  W ) ( R  .x.  ( N `  X )
) )  =  ( 0g `  W ) ) )
2822, 27mpbird 223 . 2  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( N `  ( R  .x.  X ) )  =  ( R  .x.  ( N `  X )
) )
2928eqcomd 2301 1  |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( N `  X ) )  =  ( N `  ( R  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   LModclmod 15643
This theorem is referenced by:  invlmhm  15815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-lmod 15645
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