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Theorem lmodvneg1 15942
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 15914 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Grp )
43adantr 452 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
5 eqid 2404 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmodvneg1.u . . . . . . 7  |-  .1.  =  ( 1r `  F )
72, 5, 6lmod1cl 15932 . . . . . 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 14805 . . . . 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 15922 . . . 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 2404 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
18 eqid 2404 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
1913, 17, 18lmod0vrid 15936 . . 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 15940 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  X )  e.  V )
2313, 17lmodass 15920 . . . . 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 15933 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .1.  .x.  X )  =  X )
2625oveq2d 6056 . . . . . 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 2404 . . . . . . . . . 10  |-  ( +g  `  F )  =  ( +g  `  F )
28 eqid 2404 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
295, 27, 28, 9grplinv 14806 . . . . . . . . 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 6055 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( 0g `  F
)  .x.  X )
)
3213, 17, 2, 14, 5, 27lmodvsdir 15929 . . . . . . . 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 15938 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  F
)  .x.  X )  =  ( 0g `  W ) )
3531, 33, 343eqtr3d 2444 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( 0g `  W
) )
3626, 35eqtr3d 2438 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) X )  =  ( 0g `  W
) )
3736oveq1d 6055 . . . 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 2438 . . 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 15941 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
4039oveq2d 6056 . . 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 15935 . . . 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 2444 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( N `  X
) )
4420, 43eqtr3d 2438 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 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641   1rcur 15617   LModclmod 15905
This theorem is referenced by:  lmodvsneg  15943  lmodvsubval2  15954  lssvnegcl  15987  lspsnneg  16037  lmodvsinv  16067  lspsolvlem  16169  tlmtgp  18178  clmvneg1  19069  deg1invg  19982
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907
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