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Theorem lcdvsubval 31808
Description: The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
Hypotheses
Ref Expression
lcdvsubval.h  |-  H  =  ( LHyp `  K
)
lcdvsubval.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcdvsubval.v  |-  V  =  ( Base `  U
)
lcdvsubval.r  |-  R  =  (Scalar `  U )
lcdvsubval.s  |-  S  =  ( -g `  R
)
lcdvsubval.c  |-  C  =  ( (LCDual `  K
) `  W )
lcdvsubval.d  |-  D  =  ( Base `  C
)
lcdvsubval.m  |-  .-  =  ( -g `  C )
lcdvsubval.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcdvsubval.f  |-  ( ph  ->  F  e.  D )
lcdvsubval.g  |-  ( ph  ->  G  e.  D )
lcdvsubval.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lcdvsubval  |-  ( ph  ->  ( ( F  .-  G ) `  X
)  =  ( ( F `  X ) S ( G `  X ) ) )

Proof of Theorem lcdvsubval
StepHypRef Expression
1 lcdvsubval.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 lcdvsubval.c . . . . 5  |-  C  =  ( (LCDual `  K
) `  W )
3 lcdvsubval.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 31782 . . . 4  |-  ( ph  ->  C  e.  LMod )
5 lcdvsubval.f . . . 4  |-  ( ph  ->  F  e.  D )
6 lcdvsubval.g . . . 4  |-  ( ph  ->  G  e.  D )
7 lcdvsubval.d . . . . 5  |-  D  =  ( Base `  C
)
8 eqid 2283 . . . . 5  |-  ( +g  `  C )  =  ( +g  `  C )
9 lcdvsubval.m . . . . 5  |-  .-  =  ( -g `  C )
10 eqid 2283 . . . . 5  |-  (Scalar `  C )  =  (Scalar `  C )
11 eqid 2283 . . . . 5  |-  ( .s
`  C )  =  ( .s `  C
)
12 eqid 2283 . . . . 5  |-  ( inv g `  (Scalar `  C ) )  =  ( inv g `  (Scalar `  C ) )
13 eqid 2283 . . . . 5  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
147, 8, 9, 10, 11, 12, 13lmodvsubval2 15680 . . . 4  |-  ( ( C  e.  LMod  /\  F  e.  D  /\  G  e.  D )  ->  ( F  .-  G )  =  ( F ( +g  `  C ) ( ( ( inv g `  (Scalar `  C ) ) `
 ( 1r `  (Scalar `  C ) ) ) ( .s `  C ) G ) ) )
154, 5, 6, 14syl3anc 1182 . . 3  |-  ( ph  ->  ( F  .-  G
)  =  ( F ( +g  `  C
) ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) ) )
1615fveq1d 5527 . 2  |-  ( ph  ->  ( ( F  .-  G ) `  X
)  =  ( ( F ( +g  `  C
) ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) ) `
 X ) )
17 lcdvsubval.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
18 lcdvsubval.v . . 3  |-  V  =  ( Base `  U
)
19 lcdvsubval.r . . 3  |-  R  =  (Scalar `  U )
20 eqid 2283 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
21 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2210lmodfgrp 15636 . . . . . . 7  |-  ( C  e.  LMod  ->  (Scalar `  C )  e.  Grp )
234, 22syl 15 . . . . . 6  |-  ( ph  ->  (Scalar `  C )  e.  Grp )
2410lmodrng 15635 . . . . . . . 8  |-  ( C  e.  LMod  ->  (Scalar `  C )  e.  Ring )
254, 24syl 15 . . . . . . 7  |-  ( ph  ->  (Scalar `  C )  e.  Ring )
26 eqid 2283 . . . . . . . 8  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
2726, 13rngidcl 15361 . . . . . . 7  |-  ( (Scalar `  C )  e.  Ring  -> 
( 1r `  (Scalar `  C ) )  e.  ( Base `  (Scalar `  C ) ) )
2825, 27syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  e.  ( Base `  (Scalar `  C ) ) )
2926, 12grpinvcl 14527 . . . . . 6  |-  ( ( (Scalar `  C )  e.  Grp  /\  ( 1r
`  (Scalar `  C )
)  e.  ( Base `  (Scalar `  C )
) )  ->  (
( inv g `  (Scalar `  C ) ) `
 ( 1r `  (Scalar `  C ) ) )  e.  ( Base `  (Scalar `  C )
) )
3023, 28, 29syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  e.  (
Base `  (Scalar `  C
) ) )
311, 17, 19, 21, 2, 10, 26, 3lcdsbase 31790 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  ( Base `  R
) )
3230, 31eleqtrd 2359 . . . 4  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  e.  (
Base `  R )
)
331, 17, 19, 21, 2, 7, 11, 3, 32, 6lcdvscl 31795 . . 3  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G )  e.  D )
34 lcdvsubval.x . . 3  |-  ( ph  ->  X  e.  V )
351, 17, 18, 19, 20, 2, 7, 8, 3, 5, 33, 34lcdvaddval 31788 . 2  |-  ( ph  ->  ( ( F ( +g  `  C ) ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) ) `
 X )  =  ( ( F `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) ) ( .s
`  C ) G ) `  X ) ) )
36 eqid 2283 . . . . . . . . 9  |-  ( inv g `  R )  =  ( inv g `  R )
371, 17, 19, 36, 2, 10, 12, 3lcdneg 31800 . . . . . . . 8  |-  ( ph  ->  ( inv g `  (Scalar `  C ) )  =  ( inv g `  R ) )
38 eqid 2283 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
391, 17, 19, 38, 2, 10, 13, 3lcd1 31799 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  R
) )
4037, 39fveq12d 5531 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  =  ( ( inv g `  R ) `  ( 1r `  R ) ) )
4140oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G )  =  ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  C ) G ) )
4241fveq1d 5527 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) `  X )  =  ( ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  C ) G ) `  X
) )
43 eqid 2283 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
441, 17, 3dvhlmod 31300 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
4519lmodrng 15635 . . . . . . . . 9  |-  ( U  e.  LMod  ->  R  e. 
Ring )
4644, 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 )
4919, 21, 38lmod1cl 15657 . . . . . . . 8  |-  ( U  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
5044, 49syl 15 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5121, 36grpinvcl 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
) )
531, 17, 18, 19, 21, 43, 2, 7, 11, 3, 52, 6, 34lcdvsval 31794 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  R
) `  ( 1r `  R ) ) ( .s `  C ) G ) `  X
)  =  ( ( G `  X ) ( .r `  R
) ( ( inv g `  R ) `
 ( 1r `  R ) ) ) )
541, 17, 18, 19, 21, 2, 7, 3, 6, 34lcdvbasecl 31786 . . . . . 6  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
5521, 43, 38, 36, 46, 54rngnegr 15381 . . . . 5  |-  ( ph  ->  ( ( G `  X ) ( .r
`  R ) ( ( inv g `  R ) `  ( 1r `  R ) ) )  =  ( ( inv g `  R
) `  ( G `  X ) ) )
5642, 53, 553eqtrd 2319 . . . 4  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) `  X )  =  ( ( inv g `  R ) `  ( G `  X )
) )
5756oveq2d 5874 . . 3  |-  ( ph  ->  ( ( F `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) ) ( .s
`  C ) G ) `  X ) )  =  ( ( F `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( G `  X ) ) ) )
581, 17, 18, 19, 21, 2, 7, 3, 5, 34lcdvbasecl 31786 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  R ) )
59 lcdvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6021, 20, 36, 59grpsubval 14525 . . . 4  |-  ( ( ( F `  X
)  e.  ( Base `  R )  /\  ( G `  X )  e.  ( Base `  R
) )  ->  (
( F `  X
) S ( G `
 X ) )  =  ( ( F `
 X ) ( +g  `  R ) ( ( inv g `  R ) `  ( G `  X )
) ) )
6158, 54, 60syl2anc 642 . . 3  |-  ( ph  ->  ( ( F `  X ) S ( G `  X ) )  =  ( ( F `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( G `  X ) ) ) )
6257, 61eqtr4d 2318 . 2  |-  ( ph  ->  ( ( F `  X ) ( +g  `  R ) ( ( ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) ) ( .s
`  C ) G ) `  X ) )  =  ( ( F `  X ) S ( G `  X ) ) )
6316, 35, 623eqtrd 2319 1  |-  ( ph  ->  ( ( F  .-  G ) `  X
)  =  ( ( F `  X ) S ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776
This theorem is referenced by:  hdmapinvlem3  32113
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-fal 1311  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-iin 3908  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-undef 6298  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-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777
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