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Theorem ldualvsubval 34605
Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 34603? (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 34601 . . . 4  |-  ( ph  ->  D  e.  LMod )
4 ldualvsubval.f . . . . 5  |-  F  =  (LFnl `  W )
5 eqid 2441 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
6 ldualvsubval.g . . . . 5  |-  ( ph  ->  G  e.  F )
74, 1, 5, 2, 6ldualelvbase 34575 . . . 4  |-  ( ph  ->  G  e.  ( Base `  D ) )
8 ldualvsubval.h . . . . 5  |-  ( ph  ->  H  e.  F )
94, 1, 5, 2, 8ldualelvbase 34575 . . . 4  |-  ( ph  ->  H  e.  ( Base `  D ) )
10 eqid 2441 . . . . 5  |-  ( +g  `  D )  =  ( +g  `  D )
11 ldualvsubval.m . . . . 5  |-  .-  =  ( -g `  D )
12 eqid 2441 . . . . 5  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2441 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
14 eqid 2441 . . . . 5  |-  ( invg `  (Scalar `  D ) )  =  ( invg `  (Scalar `  D ) )
15 eqid 2441 . . . . 5  |-  ( 1r
`  (Scalar `  D )
)  =  ( 1r
`  (Scalar `  D )
)
165, 10, 11, 12, 13, 14, 15lmodvsubval2 17436 . . . 4  |-  ( ( D  e.  LMod  /\  G  e.  ( Base `  D
)  /\  H  e.  ( Base `  D )
)  ->  ( G  .-  H )  =  ( G ( +g  `  D
) ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
173, 7, 9, 16syl3anc 1227 . . 3  |-  ( ph  ->  ( G  .-  H
)  =  ( G ( +g  `  D
) ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
1817fveq1d 5855 . 2  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G ( +g  `  D
) ( ( ( invg `  (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 2441 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
22 eqid 2441 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2312lmodfgrp 17392 . . . . . . 7  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Grp )
243, 23syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  D )  e.  Grp )
2512lmodring 17391 . . . . . . . 8  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Ring )
263, 25syl 16 . . . . . . 7  |-  ( ph  ->  (Scalar `  D )  e.  Ring )
27 eqid 2441 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
2827, 15ringidcl 17090 . . . . . . 7  |-  ( (Scalar `  D )  e.  Ring  -> 
( 1r `  (Scalar `  D ) )  e.  ( Base `  (Scalar `  D ) ) )
2926, 28syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  e.  ( Base `  (Scalar `  D ) ) )
3027, 14grpinvcl 15966 . . . . . 6  |-  ( ( (Scalar `  D )  e.  Grp  /\  ( 1r
`  (Scalar `  D )
)  e.  ( Base `  (Scalar `  D )
) )  ->  (
( invg `  (Scalar `  D ) ) `
 ( 1r `  (Scalar `  D ) ) )  e.  ( Base `  (Scalar `  D )
) )
3124, 29, 30syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  (Scalar `  D
) ) )
3220, 22, 1, 12, 27, 2ldualsbase 34581 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  R
) )
3331, 32eleqtrd 2531 . . . 4  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  R )
)
344, 20, 22, 1, 13, 2, 33, 8ldualvscl 34587 . . 3  |-  ( ph  ->  ( ( ( invg `  (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 34579 . 2  |-  ( ph  ->  ( ( G ( +g  `  D ) ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) `
 X )  =  ( ( G `  X ) ( +g  `  R ) ( ( ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) ) )
37 eqid 2441 . . . . . . . . 9  |-  ( invg `  R )  =  ( invg `  R )
3820, 37, 1, 12, 14, 2ldualneg 34597 . . . . . . . 8  |-  ( ph  ->  ( invg `  (Scalar `  D ) )  =  ( invg `  R ) )
39 eqid 2441 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
4020, 39, 1, 12, 15, 2ldual1 34596 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  =  ( 1r `  R
) )
4138, 40fveq12d 5859 . . . . . . 7  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
4241oveq1d 6293 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  =  ( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) )
4342fveq1d 5855 . . . . 5  |-  ( ph  ->  ( ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) `  X
) )
44 eqid 2441 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4520lmodring 17391 . . . . . . . . 9  |-  ( W  e.  LMod  ->  R  e. 
Ring )
462, 45syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 ringgrp 17074 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4920, 22, 39lmod1cl 17410 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
502, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5122, 37grpinvcl 15966 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  (
( invg `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
5248, 50, 51syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( invg `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
534, 19, 20, 22, 44, 1, 13, 2, 52, 8, 35ldualvsval 34586 . . . . 5  |-  ( ph  ->  ( ( ( ( invg `  R
) `  ( 1r `  R ) ) ( .s `  D ) H ) `  X
)  =  ( ( H `  X ) ( .r `  R
) ( ( invg `  R ) `
 ( 1r `  R ) ) ) )
5420, 22, 19, 4lflcl 34512 . . . . . . 7  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  X  e.  V )  ->  ( H `  X )  e.  ( Base `  R
) )
552, 8, 35, 54syl3anc 1227 . . . . . 6  |-  ( ph  ->  ( H `  X
)  e.  ( Base `  R ) )
5622, 44, 39, 37, 46, 55rngnegr 17112 . . . . 5  |-  ( ph  ->  ( ( H `  X ) ( .r
`  R ) ( ( invg `  R ) `  ( 1r `  R ) ) )  =  ( ( invg `  R
) `  ( H `  X ) ) )
5743, 53, 563eqtrd 2486 . . . 4  |-  ( ph  ->  ( ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( invg `  R ) `  ( H `  X )
) )
5857oveq2d 6294 . . 3  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( invg `  R ) `
 ( H `  X ) ) ) )
5920, 22, 19, 4lflcl 34512 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  R
) )
602, 6, 35, 59syl3anc 1227 . . . 4  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
61 ldualvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6222, 21, 37, 61grpsubval 15964 . . . 4  |-  ( ( ( G `  X
)  e.  ( Base `  R )  /\  ( H `  X )  e.  ( Base `  R
) )  ->  (
( G `  X
) S ( H `
 X ) )  =  ( ( G `
 X ) ( +g  `  R ) ( ( invg `  R ) `  ( H `  X )
) ) )
6360, 55, 62syl2anc 661 . . 3  |-  ( ph  ->  ( ( G `  X ) S ( H `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( invg `  R ) `
 ( H `  X ) ) ) )
6458, 63eqtr4d 2485 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) S ( H `  X ) ) )
6518, 36, 643eqtrd 2486 1  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   ` cfv 5575  (class class class)co 6278   Basecbs 14506   +g cplusg 14571   .rcmulr 14572  Scalarcsca 14574   .scvsca 14575   Grpcgrp 15924   invgcminusg 15925   -gcsg 15926   1rcur 17024   Ringcrg 17069   LModclmod 17383  LFnlclfn 34505  LDualcld 34571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6522  df-om 6683  df-1st 6782  df-2nd 6783  df-tpos 6954  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-plusg 14584  df-mulr 14585  df-sca 14587  df-vsca 14588  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-minusg 15929  df-sbg 15930  df-cmn 16671  df-abl 16672  df-mgp 17013  df-ur 17025  df-ring 17071  df-oppr 17143  df-lmod 17385  df-lfl 34506  df-ldual 34572
This theorem is referenced by:  lcfrlem1  36992
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