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Theorem ldualvsubval 29955
Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 29953? (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 29951 . . . 4  |-  ( ph  ->  D  e.  LMod )
4 ldualvsubval.f . . . . 5  |-  F  =  (LFnl `  W )
5 eqid 2436 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
6 ldualvsubval.g . . . . 5  |-  ( ph  ->  G  e.  F )
74, 1, 5, 2, 6ldualelvbase 29925 . . . 4  |-  ( ph  ->  G  e.  ( Base `  D ) )
8 ldualvsubval.h . . . . 5  |-  ( ph  ->  H  e.  F )
94, 1, 5, 2, 8ldualelvbase 29925 . . . 4  |-  ( ph  ->  H  e.  ( Base `  D ) )
10 eqid 2436 . . . . 5  |-  ( +g  `  D )  =  ( +g  `  D )
11 ldualvsubval.m . . . . 5  |-  .-  =  ( -g `  D )
12 eqid 2436 . . . . 5  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2436 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
14 eqid 2436 . . . . 5  |-  ( inv g `  (Scalar `  D ) )  =  ( inv g `  (Scalar `  D ) )
15 eqid 2436 . . . . 5  |-  ( 1r
`  (Scalar `  D )
)  =  ( 1r
`  (Scalar `  D )
)
165, 10, 11, 12, 13, 14, 15lmodvsubval2 15999 . . . 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 1184 . . 3  |-  ( ph  ->  ( G  .-  H
)  =  ( G ( +g  `  D
) ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
1817fveq1d 5730 . 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 2436 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
22 eqid 2436 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2312lmodfgrp 15959 . . . . . . 7  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Grp )
243, 23syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  D )  e.  Grp )
2512lmodrng 15958 . . . . . . . 8  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Ring )
263, 25syl 16 . . . . . . 7  |-  ( ph  ->  (Scalar `  D )  e.  Ring )
27 eqid 2436 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
2827, 15rngidcl 15684 . . . . . . 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 14850 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  (Scalar `  D
) ) )
3220, 22, 1, 12, 27, 2ldualsbase 29931 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  R
) )
3331, 32eleqtrd 2512 . . . 4  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  R )
)
344, 20, 22, 1, 13, 2, 33, 8ldualvscl 29937 . . 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 29929 . 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 2436 . . . . . . . . 9  |-  ( inv g `  R )  =  ( inv g `  R )
3820, 37, 1, 12, 14, 2ldualneg 29947 . . . . . . . 8  |-  ( ph  ->  ( inv g `  (Scalar `  D ) )  =  ( inv g `  R ) )
39 eqid 2436 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
4020, 39, 1, 12, 15, 2ldual1 29946 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  =  ( 1r `  R
) )
4138, 40fveq12d 5734 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  =  ( ( inv g `  R ) `  ( 1r `  R ) ) )
4241oveq1d 6096 . . . . . 6  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  =  ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) )
4342fveq1d 5730 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) `  X
) )
44 eqid 2436 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4520lmodrng 15958 . . . . . . . . 9  |-  ( W  e.  LMod  ->  R  e. 
Ring )
462, 45syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 rnggrp 15669 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4920, 22, 39lmod1cl 15977 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
502, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5122, 37grpinvcl 14850 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  (
( inv g `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
5248, 50, 51syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( inv g `  R ) `  ( 1r `  R ) )  e.  ( Base `  R
) )
534, 19, 20, 22, 44, 1, 13, 2, 52, 8, 35ldualvsval 29936 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  R
) `  ( 1r `  R ) ) ( .s `  D ) H ) `  X
)  =  ( ( H `  X ) ( .r `  R
) ( ( inv g `  R ) `
 ( 1r `  R ) ) ) )
5420, 22, 19, 4lflcl 29862 . . . . . . 7  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  X  e.  V )  ->  ( H `  X )  e.  ( Base `  R
) )
552, 8, 35, 54syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( H `  X
)  e.  ( Base `  R ) )
5622, 44, 39, 37, 46, 55rngnegr 15704 . . . . 5  |-  ( ph  ->  ( ( H `  X ) ( .r
`  R ) ( ( inv g `  R ) `  ( 1r `  R ) ) )  =  ( ( inv g `  R
) `  ( H `  X ) ) )
5743, 53, 563eqtrd 2472 . . . 4  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( inv g `  R ) `  ( H `  X )
) )
5857oveq2d 6097 . . 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 29862 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  R
) )
602, 6, 35, 59syl3anc 1184 . . . 4  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
61 ldualvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6222, 21, 37, 61grpsubval 14848 . . . 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 643 . . 3  |-  ( ph  ->  ( ( G `  X ) S ( H `  X ) )  =  ( ( G `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( H `  X ) ) ) )
6458, 63eqtr4d 2471 . 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 2472 1  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688   Ringcrg 15660   1rcur 15662   LModclmod 15950  LFnlclfn 29855  LDualcld 29921
This theorem is referenced by:  lcfrlem1  32340
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-int 4051  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-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  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-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-lmod 15952  df-lfl 29856  df-ldual 29922
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