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Theorem lmodvneg1 15987
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 15959 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Grp )
43adantr 452 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
5 eqid 2436 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmodvneg1.u . . . . . . 7  |-  .1.  =  ( 1r `  F )
72, 5, 6lmod1cl 15977 . . . . . 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 14850 . . . . 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 15967 . . . 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 2436 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
18 eqid 2436 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
1913, 17, 18lmod0vrid 15981 . . 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 15985 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  X )  e.  V )
2313, 17lmodass 15965 . . . . 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 15978 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .1.  .x.  X )  =  X )
2625oveq2d 6097 . . . . . 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 2436 . . . . . . . . . 10  |-  ( +g  `  F )  =  ( +g  `  F )
28 eqid 2436 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
295, 27, 28, 9grplinv 14851 . . . . . . . . 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 6096 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( 0g `  F
)  .x.  X )
)
3213, 17, 2, 14, 5, 27lmodvsdir 15974 . . . . . . . 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 15983 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  F
)  .x.  X )  =  ( 0g `  W ) )
3531, 33, 343eqtr3d 2476 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( 0g `  W
) )
3626, 35eqtr3d 2470 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) X )  =  ( 0g `  W
) )
3736oveq1d 6096 . . . 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 2470 . . 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 15986 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
4039oveq2d 6097 . . 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 15980 . . . 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 2476 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( N `  X
) )
4420, 43eqtr3d 2470 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 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686   1rcur 15662   LModclmod 15950
This theorem is referenced by:  lmodvsneg  15988  lmodvsubval2  15999  lssvnegcl  16032  lspsnneg  16082  lmodvsinv  16112  lspsolvlem  16214  tlmtgp  18225  clmvneg1  19116  deg1invg  20029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952
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