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Theorem lmodvneg1 16968
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  =  ( invg `  W )
lmodvneg1.f  |-  F  =  (Scalar `  W )
lmodvneg1.s  |-  .x.  =  ( .s `  W )
lmodvneg1.u  |-  .1.  =  ( 1r `  F )
lmodvneg1.m  |-  M  =  ( invg `  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 454 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmodvneg1.f . . . . . . 7  |-  F  =  (Scalar `  W )
32lmodfgrp 16937 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Grp )
43adantr 462 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
5 eqid 2441 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmodvneg1.u . . . . . . 7  |-  .1.  =  ( 1r `  F )
72, 5, 6lmod1cl 16955 . . . . . 6  |-  ( W  e.  LMod  ->  .1.  e.  ( Base `  F )
)
87adantr 462 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .1.  e.  ( Base `  F
) )
9 lmodvneg1.m . . . . . 6  |-  M  =  ( invg `  F )
105, 9grpinvcl 15576 . . . . 5  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
114, 8, 10syl2anc 656 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
12 simpr 458 . . . 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 16945 . . . 4  |-  ( ( W  e.  LMod  /\  ( M `  .1.  )  e.  ( Base `  F
)  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
161, 11, 12, 15syl3anc 1213 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
17 eqid 2441 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
18 eqid 2441 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
1913, 17, 18lmod0vrid 16959 . . 3  |-  ( ( W  e.  LMod  /\  (
( M `  .1.  )  .x.  X )  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( ( M `  .1.  )  .x.  X ) )
2016, 19syldan 467 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( ( M `  .1.  )  .x.  X ) )
21 lmodvneg1.n . . . . . 6  |-  N  =  ( invg `  W )
2213, 21lmodvnegcl 16966 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  X )  e.  V )
2313, 17lmodass 16943 . . . . 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 1215 . . . 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 16956 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .1.  .x.  X )  =  X )
2625oveq2d 6106 . . . . . 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 2441 . . . . . . . . . 10  |-  ( +g  `  F )  =  ( +g  `  F )
28 eqid 2441 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
295, 27, 28, 9grplinv 15577 . . . . . . . . 9  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
304, 8, 29syl2anc 656 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
3130oveq1d 6105 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( 0g `  F
)  .x.  X )
)
3213, 17, 2, 14, 5, 27lmodvsdir 16952 . . . . . . . 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 1215 . . . . . . 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 16961 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  F
)  .x.  X )  =  ( 0g `  W ) )
3531, 33, 343eqtr3d 2481 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( 0g `  W
) )
3626, 35eqtr3d 2475 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) X )  =  ( 0g `  W
) )
3736oveq1d 6105 . . . 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 2475 . . 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 16967 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
4039oveq2d 6106 . . 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 16958 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  X )  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4222, 41syldan 467 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4338, 40, 423eqtr3d 2481 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( N `  X
) )
4420, 43eqtr3d 2475 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  =  ( N `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   .scvsca 14238   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407   1rcur 16593   LModclmod 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930
This theorem is referenced by:  lmodvsneg  16969  lmodvsubval2  16980  lssvnegcl  17015  lspsnneg  17065  lmodvsinv  17095  lspsolvlem  17201  tlmtgp  19729  clmvneg1  20622  deg1invg  21537
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