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Theorem ldualvsubval 29347
Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 29345? (Requires  D to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
Hypotheses
Ref Expression
ldualvsubval.v  |-  V  =  ( Base `  W
)
ldualvsubval.r  |-  R  =  (Scalar `  W )
ldualvsubval.s  |-  S  =  ( -g `  R
)
ldualvsubval.f  |-  F  =  (LFnl `  W )
ldualvsubval.d  |-  D  =  (LDual `  W )
ldualvsubval.m  |-  .-  =  ( -g `  D )
ldualvsubval.w  |-  ( ph  ->  W  e.  LMod )
ldualvsubval.g  |-  ( ph  ->  G  e.  F )
ldualvsubval.h  |-  ( ph  ->  H  e.  F )
ldualvsubval.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
ldualvsubval  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )

Proof of Theorem ldualvsubval
StepHypRef Expression
1 ldualvsubval.d . . . . 5  |-  D  =  (LDual `  W )
2 ldualvsubval.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
31, 2lduallmod 29343 . . . 4  |-  ( ph  ->  D  e.  LMod )
4 ldualvsubval.f . . . . 5  |-  F  =  (LFnl `  W )
5 eqid 2283 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
6 ldualvsubval.g . . . . 5  |-  ( ph  ->  G  e.  F )
74, 1, 5, 2, 6ldualelvbase 29317 . . . 4  |-  ( ph  ->  G  e.  ( Base `  D ) )
8 ldualvsubval.h . . . . 5  |-  ( ph  ->  H  e.  F )
94, 1, 5, 2, 8ldualelvbase 29317 . . . 4  |-  ( ph  ->  H  e.  ( Base `  D ) )
10 eqid 2283 . . . . 5  |-  ( +g  `  D )  =  ( +g  `  D )
11 ldualvsubval.m . . . . 5  |-  .-  =  ( -g `  D )
12 eqid 2283 . . . . 5  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2283 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
14 eqid 2283 . . . . 5  |-  ( inv g `  (Scalar `  D ) )  =  ( inv g `  (Scalar `  D ) )
15 eqid 2283 . . . . 5  |-  ( 1r
`  (Scalar `  D )
)  =  ( 1r
`  (Scalar `  D )
)
165, 10, 11, 12, 13, 14, 15lmodvsubval2 15680 . . . 4  |-  ( ( D  e.  LMod  /\  G  e.  ( Base `  D
)  /\  H  e.  ( Base `  D )
)  ->  ( G  .-  H )  =  ( G ( +g  `  D
) ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
173, 7, 9, 16syl3anc 1182 . . 3  |-  ( ph  ->  ( G  .-  H
)  =  ( G ( +g  `  D
) ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
1817fveq1d 5527 . 2  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G ( +g  `  D
) ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) `
 X ) )
19 ldualvsubval.v . . 3  |-  V  =  ( Base `  W
)
20 ldualvsubval.r . . 3  |-  R  =  (Scalar `  W )
21 eqid 2283 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
22 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2312lmodfgrp 15636 . . . . . . 7  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Grp )
243, 23syl 15 . . . . . 6  |-  ( ph  ->  (Scalar `  D )  e.  Grp )
2512lmodrng 15635 . . . . . . . 8  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Ring )
263, 25syl 15 . . . . . . 7  |-  ( ph  ->  (Scalar `  D )  e.  Ring )
27 eqid 2283 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
2827, 15rngidcl 15361 . . . . . . 7  |-  ( (Scalar `  D )  e.  Ring  -> 
( 1r `  (Scalar `  D ) )  e.  ( Base `  (Scalar `  D ) ) )
2926, 28syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  e.  ( Base `  (Scalar `  D ) ) )
3027, 14grpinvcl 14527 . . . . . 6  |-  ( ( (Scalar `  D )  e.  Grp  /\  ( 1r
`  (Scalar `  D )
)  e.  ( Base `  (Scalar `  D )
) )  ->  (
( inv g `  (Scalar `  D ) ) `
 ( 1r `  (Scalar `  D ) ) )  e.  ( Base `  (Scalar `  D )
) )
3124, 29, 30syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  (Scalar `  D
) ) )
3220, 22, 1, 12, 27, 2ldualsbase 29323 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  R
) )
3331, 32eleqtrd 2359 . . . 4  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  R )
)
344, 20, 22, 1, 13, 2, 33, 8ldualvscl 29329 . . 3  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  e.  F )
35 ldualvsubval.x . . 3  |-  ( ph  ->  X  e.  V )
3619, 20, 21, 4, 1, 10, 2, 6, 34, 35ldualvaddval 29321 . 2  |-  ( ph  ->  ( ( G ( +g  `  D ) ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) `
 X )  =  ( ( G `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) ) )
37 eqid 2283 . . . . . . . . 9  |-  ( inv g `  R )  =  ( inv g `  R )
3820, 37, 1, 12, 14, 2ldualneg 29339 . . . . . . . 8  |-  ( ph  ->  ( inv g `  (Scalar `  D ) )  =  ( inv g `  R ) )
39 eqid 2283 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
4020, 39, 1, 12, 15, 2ldual1 29338 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  =  ( 1r `  R
) )
4138, 40fveq12d 5531 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  =  ( ( inv g `  R ) `  ( 1r `  R ) ) )
4241oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  =  ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) )
4342fveq1d 5527 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) `  X
) )
44 eqid 2283 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4520lmodrng 15635 . . . . . . . . 9  |-  ( W  e.  LMod  ->  R  e. 
Ring )
462, 45syl 15 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 rnggrp 15346 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 15 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4920, 22, 39lmod1cl 15657 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
502, 49syl 15 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5122, 37grpinvcl 14527 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  (
( inv g `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
5248, 50, 51syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( inv g `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
534, 19, 20, 22, 44, 1, 13, 2, 52, 8, 35ldualvsval 29328 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  R
) `  ( 1r `  R ) ) ( .s `  D ) H ) `  X
)  =  ( ( H `  X ) ( .r `  R
) ( ( inv g `  R ) `
 ( 1r `  R ) ) ) )
5420, 22, 19, 4lflcl 29254 . . . . . . 7  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  X  e.  V )  ->  ( H `  X )  e.  ( Base `  R
) )
552, 8, 35, 54syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( H `  X
)  e.  ( Base `  R ) )
5622, 44, 39, 37, 46, 55rngnegr 15381 . . . . 5  |-  ( ph  ->  ( ( H `  X ) ( .r
`  R ) ( ( inv g `  R ) `  ( 1r `  R ) ) )  =  ( ( inv g `  R
) `  ( H `  X ) ) )
5743, 53, 563eqtrd 2319 . . . 4  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( inv g `  R ) `  ( H `  X )
) )
5857oveq2d 5874 . . 3  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( H `  X ) ) ) )
5920, 22, 19, 4lflcl 29254 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  R
) )
602, 6, 35, 59syl3anc 1182 . . . 4  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
61 ldualvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6222, 21, 37, 61grpsubval 14525 . . . 4  |-  ( ( ( G `  X
)  e.  ( Base `  R )  /\  ( H `  X )  e.  ( Base `  R
) )  ->  (
( G `  X
) S ( H `
 X ) )  =  ( ( G `
 X ) ( +g  `  R ) ( ( inv g `  R ) `  ( H `  X )
) ) )
6360, 55, 62syl2anc 642 . . 3  |-  ( ph  ->  ( ( G `  X ) S ( H `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( H `  X ) ) ) )
6458, 63eqtr4d 2318 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) S ( H `  X ) ) )
6518, 36, 643eqtrd 2319 1  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   Ringcrg 15337   1rcur 15339   LModclmod 15627  LFnlclfn 29247  LDualcld 29313
This theorem is referenced by:  lcfrlem1  31732
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-int 3863  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-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-lmod 15629  df-lfl 29248  df-ldual 29314
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