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Theorem ldualvsubval 32898
Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 32896? (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 32894 . . . 4  |-  ( ph  ->  D  e.  LMod )
4 ldualvsubval.f . . . . 5  |-  F  =  (LFnl `  W )
5 eqid 2443 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
6 ldualvsubval.g . . . . 5  |-  ( ph  ->  G  e.  F )
74, 1, 5, 2, 6ldualelvbase 32868 . . . 4  |-  ( ph  ->  G  e.  ( Base `  D ) )
8 ldualvsubval.h . . . . 5  |-  ( ph  ->  H  e.  F )
94, 1, 5, 2, 8ldualelvbase 32868 . . . 4  |-  ( ph  ->  H  e.  ( Base `  D ) )
10 eqid 2443 . . . . 5  |-  ( +g  `  D )  =  ( +g  `  D )
11 ldualvsubval.m . . . . 5  |-  .-  =  ( -g `  D )
12 eqid 2443 . . . . 5  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2443 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
14 eqid 2443 . . . . 5  |-  ( invg `  (Scalar `  D ) )  =  ( invg `  (Scalar `  D ) )
15 eqid 2443 . . . . 5  |-  ( 1r
`  (Scalar `  D )
)  =  ( 1r
`  (Scalar `  D )
)
165, 10, 11, 12, 13, 14, 15lmodvsubval2 17022 . . . 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 1218 . . 3  |-  ( ph  ->  ( G  .-  H
)  =  ( G ( +g  `  D
) ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) ) )
1817fveq1d 5714 . 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 2443 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
22 eqid 2443 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2312lmodfgrp 16979 . . . . . . 7  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Grp )
243, 23syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  D )  e.  Grp )
2512lmodrng 16978 . . . . . . . 8  |-  ( D  e.  LMod  ->  (Scalar `  D )  e.  Ring )
263, 25syl 16 . . . . . . 7  |-  ( ph  ->  (Scalar `  D )  e.  Ring )
27 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
2827, 15rngidcl 16687 . . . . . . 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 15604 . . . . . 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 32874 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  R
) )
3331, 32eleqtrd 2519 . . . 4  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  e.  (
Base `  R )
)
344, 20, 22, 1, 13, 2, 33, 8ldualvscl 32880 . . 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 32872 . 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 2443 . . . . . . . . 9  |-  ( invg `  R )  =  ( invg `  R )
3820, 37, 1, 12, 14, 2ldualneg 32890 . . . . . . . 8  |-  ( ph  ->  ( invg `  (Scalar `  D ) )  =  ( invg `  R ) )
39 eqid 2443 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
4020, 39, 1, 12, 15, 2ldual1 32889 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  D ) )  =  ( 1r `  R
) )
4138, 40fveq12d 5718 . . . . . . 7  |-  ( ph  ->  ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
4241oveq1d 6127 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H )  =  ( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) )
4342fveq1d 5714 . . . . 5  |-  ( ph  ->  ( ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .s `  D ) H ) `  X
) )
44 eqid 2443 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4520lmodrng 16978 . . . . . . . . 9  |-  ( W  e.  LMod  ->  R  e. 
Ring )
462, 45syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 rnggrp 16672 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4920, 22, 39lmod1cl 16997 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
502, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5122, 37grpinvcl 15604 . . . . . . 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 32879 . . . . 5  |-  ( ph  ->  ( ( ( ( invg `  R
) `  ( 1r `  R ) ) ( .s `  D ) H ) `  X
)  =  ( ( H `  X ) ( .r `  R
) ( ( invg `  R ) `
 ( 1r `  R ) ) ) )
5420, 22, 19, 4lflcl 32805 . . . . . . 7  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  X  e.  V )  ->  ( H `  X )  e.  ( Base `  R
) )
552, 8, 35, 54syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( H `  X
)  e.  ( Base `  R ) )
5622, 44, 39, 37, 46, 55rngnegr 16708 . . . . 5  |-  ( ph  ->  ( ( H `  X ) ( .r
`  R ) ( ( invg `  R ) `  ( 1r `  R ) ) )  =  ( ( invg `  R
) `  ( H `  X ) ) )
5743, 53, 563eqtrd 2479 . . . 4  |-  ( ph  ->  ( ( ( ( invg `  (Scalar `  D ) ) `  ( 1r `  (Scalar `  D ) ) ) ( .s `  D
) H ) `  X )  =  ( ( invg `  R ) `  ( H `  X )
) )
5857oveq2d 6128 . . 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 32805 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  R
) )
602, 6, 35, 59syl3anc 1218 . . . 4  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
61 ldualvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6222, 21, 37, 61grpsubval 15602 . . . 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 2478 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  R ) ( ( ( ( invg `  (Scalar `  D )
) `  ( 1r `  (Scalar `  D )
) ) ( .s
`  D ) H ) `  X ) )  =  ( ( G `  X ) S ( H `  X ) ) )
6518, 36, 643eqtrd 2479 1  |-  ( ph  ->  ( ( G  .-  H ) `  X
)  =  ( ( G `  X ) S ( H `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   .rcmulr 14260  Scalarcsca 14262   .scvsca 14263   Grpcgrp 15431   invgcminusg 15432   -gcsg 15434   1rcur 16625   Ringcrg 16667   LModclmod 16970  LFnlclfn 32798  LDualcld 32864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-lmod 16972  df-lfl 32799  df-ldual 32865
This theorem is referenced by:  lcfrlem1  35283
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