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Theorem lmodvneg1 15946
Description: Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmodvneg1.v  |-  V  =  ( Base `  W
)
lmodvneg1.n  |-  N  =  ( inv g `  W )
lmodvneg1.f  |-  F  =  (Scalar `  W )
lmodvneg1.s  |-  .x.  =  ( .s `  W )
lmodvneg1.u  |-  .1.  =  ( 1r `  F )
lmodvneg1.m  |-  M  =  ( inv g `  F )
Assertion
Ref Expression
lmodvneg1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  =  ( N `  X
) )

Proof of Theorem lmodvneg1
StepHypRef Expression
1 simpl 444 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmodvneg1.f . . . . . . 7  |-  F  =  (Scalar `  W )
32lmodfgrp 15918 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Grp )
43adantr 452 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
5 eqid 2408 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmodvneg1.u . . . . . . 7  |-  .1.  =  ( 1r `  F )
72, 5, 6lmod1cl 15936 . . . . . 6  |-  ( W  e.  LMod  ->  .1.  e.  ( Base `  F )
)
87adantr 452 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .1.  e.  ( Base `  F
) )
9 lmodvneg1.m . . . . . 6  |-  M  =  ( inv g `  F )
105, 9grpinvcl 14809 . . . . 5  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
114, 8, 10syl2anc 643 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
12 simpr 448 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  V )
13 lmodvneg1.v . . . . 5  |-  V  =  ( Base `  W
)
14 lmodvneg1.s . . . . 5  |-  .x.  =  ( .s `  W )
1513, 2, 14, 5lmodvscl 15926 . . . 4  |-  ( ( W  e.  LMod  /\  ( M `  .1.  )  e.  ( Base `  F
)  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
161, 11, 12, 15syl3anc 1184 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
17 eqid 2408 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
18 eqid 2408 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
1913, 17, 18lmod0vrid 15940 . . 3  |-  ( ( W  e.  LMod  /\  (
( M `  .1.  )  .x.  X )  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( ( M `  .1.  )  .x.  X ) )
2016, 19syldan 457 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( ( M `  .1.  )  .x.  X ) )
21 lmodvneg1.n . . . . . 6  |-  N  =  ( inv g `  W )
2213, 21lmodvnegcl 15944 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  X )  e.  V )
2313, 17lmodass 15924 . . . . 5  |-  ( ( W  e.  LMod  /\  (
( ( M `  .1.  )  .x.  X )  e.  V  /\  X  e.  V  /\  ( N `  X )  e.  V ) )  -> 
( ( ( ( M `  .1.  )  .x.  X ) ( +g  `  W ) X ) ( +g  `  W
) ( N `  X ) )  =  ( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) ( X ( +g  `  W
) ( N `  X ) ) ) )
241, 16, 12, 22, 23syl13anc 1186 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) X ) ( +g  `  W
) ( N `  X ) )  =  ( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) ( X ( +g  `  W
) ( N `  X ) ) ) )
2513, 2, 14, 6lmodvs1 15937 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .1.  .x.  X )  =  X )
2625oveq2d 6060 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) X ) )
27 eqid 2408 . . . . . . . . . 10  |-  ( +g  `  F )  =  ( +g  `  F )
28 eqid 2408 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
295, 27, 28, 9grplinv 14810 . . . . . . . . 9  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
304, 8, 29syl2anc 643 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
3130oveq1d 6059 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( 0g `  F
)  .x.  X )
)
3213, 17, 2, 14, 5, 27lmodvsdir 15933 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
( M `  .1.  )  e.  ( Base `  F )  /\  .1.  e.  ( Base `  F
)  /\  X  e.  V ) )  -> 
( ( ( M `
 .1.  ) ( +g  `  F )  .1.  )  .x.  X
)  =  ( ( ( M `  .1.  )  .x.  X ) ( +g  `  W ) (  .1.  .x.  X
) ) )
331, 11, 8, 12, 32syl13anc 1186 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) ) )
3413, 2, 14, 28, 18lmod0vs 15942 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  F
)  .x.  X )  =  ( 0g `  W ) )
3531, 33, 343eqtr3d 2448 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( 0g `  W
) )
3626, 35eqtr3d 2442 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) X )  =  ( 0g `  W
) )
3736oveq1d 6059 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) X ) ( +g  `  W
) ( N `  X ) )  =  ( ( 0g `  W ) ( +g  `  W ) ( N `
 X ) ) )
3824, 37eqtr3d 2442 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( X ( +g  `  W ) ( N `  X
) ) )  =  ( ( 0g `  W ) ( +g  `  W ) ( N `
 X ) ) )
3913, 17, 18, 21lmodvnegid 15945 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
4039oveq2d 6060 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( X ( +g  `  W ) ( N `  X
) ) )  =  ( ( ( M `
 .1.  )  .x.  X ) ( +g  `  W ) ( 0g
`  W ) ) )
4113, 17, 18lmod0vlid 15939 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  X )  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4222, 41syldan 457 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4338, 40, 423eqtr3d 2448 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( N `  X
) )
4420, 43eqtr3d 2442 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  =  ( N `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   Grpcgrp 14644   inv gcminusg 14645   1rcur 15621   LModclmod 15909
This theorem is referenced by:  lmodvsneg  15947  lmodvsubval2  15958  lssvnegcl  15991  lspsnneg  16041  lmodvsinv  16071  lspsolvlem  16173  tlmtgp  18182  clmvneg1  19073  deg1invg  19986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-plusg 13501  df-0g 13686  df-mnd 14649  df-grp 14771  df-minusg 14772  df-mgp 15608  df-rng 15622  df-ur 15624  df-lmod 15911
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