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Theorem ldualvsubval 32524
Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 32522? (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 32520 . . . 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 32494 . . . 4  |-  ( ph  ->  G  e.  ( Base `  D ) )
8 ldualvsubval.h . . . . 5  |-  ( ph  ->  H  e.  F )
94, 1, 5, 2, 8ldualelvbase 32494 . . . 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 16980 . . . 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 1213 . . 3  |-  ( ph  ->  ( G  .-  H
)  =  ( G ( +g  `  D
) ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
1817fveq1d 5690 . 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 16937 . . . . . . 7  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Grp )
243, 23syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  D )  e.  Grp )
2512lmodrng 16936 . . . . . . . 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, 15rngidcl 16655 . . . . . . 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 15576 . . . . . 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 656 . . . . 5  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  (Scalar `  D
) ) )
3220, 22, 1, 12, 27, 2ldualsbase 32500 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  R
) )
3331, 32eleqtrd 2517 . . . 4  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  R )
)
344, 20, 22, 1, 13, 2, 33, 8ldualvscl 32506 . . 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 32498 . 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 32516 . . . . . . . 8  |-  ( ph  ->  ( invg `  (Scalar `  D ) )  =  ( invg `  R ) )
39 eqid 2441 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
4020, 39, 1, 12, 15, 2ldual1 32515 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  =  ( 1r `  R
) )
4138, 40fveq12d 5694 . . . . . . 7  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
4241oveq1d 6105 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  =  ( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) )
4342fveq1d 5690 . . . . 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
)
4520lmodrng 16936 . . . . . . . . 9  |-  ( W  e.  LMod  ->  R  e. 
Ring )
462, 45syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 rnggrp 16640 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4920, 22, 39lmod1cl 16955 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
502, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5122, 37grpinvcl 15576 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  (
( invg `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
5248, 50, 51syl2anc 656 . . . . . 6  |-  ( ph  ->  ( ( invg `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
534, 19, 20, 22, 44, 1, 13, 2, 52, 8, 35ldualvsval 32505 . . . . 5  |-  ( ph  ->  ( ( ( ( invg `  R
) `  ( 1r `  R ) ) ( .s `  D ) H ) `  X
)  =  ( ( H `  X ) ( .r `  R
) ( ( invg `  R ) `
 ( 1r `  R ) ) ) )
5420, 22, 19, 4lflcl 32431 . . . . . . 7  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  X  e.  V )  ->  ( H `  X )  e.  ( Base `  R
) )
552, 8, 35, 54syl3anc 1213 . . . . . 6  |-  ( ph  ->  ( H `  X
)  e.  ( Base `  R ) )
5622, 44, 39, 37, 46, 55rngnegr 16676 . . . . 5  |-  ( ph  ->  ( ( H `  X ) ( .r
`  R ) ( ( invg `  R ) `  ( 1r `  R ) ) )  =  ( ( invg `  R
) `  ( H `  X ) ) )
5743, 53, 563eqtrd 2477 . . . 4  |-  ( ph  ->  ( ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( invg `  R ) `  ( H `  X )
) )
5857oveq2d 6106 . . 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 32431 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  R
) )
602, 6, 35, 59syl3anc 1213 . . . 4  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
61 ldualvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6222, 21, 37, 61grpsubval 15574 . . . 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 656 . . 3  |-  ( ph  ->  ( ( G `  X ) S ( H `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( invg `  R ) `
 ( H `  X ) ) ) )
6458, 63eqtr4d 2476 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) S ( H `  X ) ) )
6518, 36, 643eqtrd 2477 1  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   .rcmulr 14235  Scalarcsca 14237   .scvsca 14238   Grpcgrp 15406   invgcminusg 15407   -gcsg 15409   1rcur 16593   Ringcrg 16635   LModclmod 16928  LFnlclfn 32424  LDualcld 32490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-lmod 16930  df-lfl 32425  df-ldual 32491
This theorem is referenced by:  lcfrlem1  34909
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