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Theorem lmodvneg1 15667
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 443 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmodvneg1.f . . . . . . 7  |-  F  =  (Scalar `  W )
32lmodfgrp 15636 . . . . . 6  |-  ( W  e.  LMod  ->  F  e. 
Grp )
43adantr 451 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
5 eqid 2283 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmodvneg1.u . . . . . . 7  |-  .1.  =  ( 1r `  F )
72, 5, 6lmod1cl 15657 . . . . . 6  |-  ( W  e.  LMod  ->  .1.  e.  ( Base `  F )
)
87adantr 451 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .1.  e.  ( Base `  F
) )
9 lmodvneg1.m . . . . . 6  |-  M  =  ( inv g `  F )
105, 9grpinvcl 14527 . . . . 5  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
114, 8, 10syl2anc 642 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( M `  .1.  )  e.  ( Base `  F
) )
12 simpr 447 . . . 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 15644 . . . 4  |-  ( ( W  e.  LMod  /\  ( M `  .1.  )  e.  ( Base `  F
)  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
161, 11, 12, 15syl3anc 1182 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  e.  V )
17 eqid 2283 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
18 eqid 2283 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
1913, 17, 18lmod0vrid 15661 . . 3  |-  ( ( W  e.  LMod  /\  (
( M `  .1.  )  .x.  X )  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( ( M `  .1.  )  .x.  X ) )
2016, 19syldan 456 . 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 15665 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  X )  e.  V )
2313, 17lmodass 15642 . . . . 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 1184 . . . 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 15658 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .1.  .x.  X )  =  X )
2625oveq2d 5874 . . . . . 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 2283 . . . . . . . . . 10  |-  ( +g  `  F )  =  ( +g  `  F )
28 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
295, 27, 28, 9grplinv 14528 . . . . . . . . 9  |-  ( ( F  e.  Grp  /\  .1.  e.  ( Base `  F
) )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
304, 8, 29syl2anc 642 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  ) ( +g  `  F
)  .1.  )  =  ( 0g `  F
) )
3130oveq1d 5873 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  ) ( +g  `  F
)  .1.  )  .x.  X )  =  ( ( 0g `  F
)  .x.  X )
)
3213, 17, 2, 14, 5, 27lmodvsdir 15652 . . . . . . . 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 1184 . . . . . . 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 15663 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  F
)  .x.  X )  =  ( 0g `  W ) )
3531, 33, 343eqtr3d 2323 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) (  .1.  .x.  X ) )  =  ( 0g `  W
) )
3626, 35eqtr3d 2317 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) X )  =  ( 0g `  W
) )
3736oveq1d 5873 . . . 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 2317 . . 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 15666 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( X ( +g  `  W
) ( N `  X ) )  =  ( 0g `  W
) )
4039oveq2d 5874 . . 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 15660 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  X )  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4222, 41syldan 456 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  W
) ( +g  `  W
) ( N `  X ) )  =  ( N `  X
) )
4338, 40, 423eqtr3d 2323 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( M `  .1.  )  .x.  X ) ( +g  `  W
) ( 0g `  W ) )  =  ( N `  X
) )
4420, 43eqtr3d 2317 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( M `  .1.  )  .x.  X )  =  ( N `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   1rcur 15339   LModclmod 15627
This theorem is referenced by:  lmodvsnegOLD  15668  lmodvsneg  15669  lmodvsubval2  15680  lssvnegcl  15713  lspsnneg  15763  lmodvsinv  15793  lspsolvlem  15895  tlmtgp  17878  clmvneg1  18589  deg1invg  19492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629
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