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Theorem lcdvsubval 31785
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 31759 . . . 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 2381 . . . . 5  |-  ( +g  `  C )  =  ( +g  `  C )
9 lcdvsubval.m . . . . 5  |-  .-  =  ( -g `  C )
10 eqid 2381 . . . . 5  |-  (Scalar `  C )  =  (Scalar `  C )
11 eqid 2381 . . . . 5  |-  ( .s
`  C )  =  ( .s `  C
)
12 eqid 2381 . . . . 5  |-  ( inv g `  (Scalar `  C ) )  =  ( inv g `  (Scalar `  C ) )
13 eqid 2381 . . . . 5  |-  ( 1r
`  (Scalar `  C )
)  =  ( 1r
`  (Scalar `  C )
)
147, 8, 9, 10, 11, 12, 13lmodvsubval2 15920 . . . 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 1184 . . 3  |-  ( ph  ->  ( F  .-  G
)  =  ( F ( +g  `  C
) ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) ) )
1615fveq1d 5664 . 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 2381 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
21 eqid 2381 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2210lmodfgrp 15880 . . . . . . 7  |-  ( C  e.  LMod  ->  (Scalar `  C )  e.  Grp )
234, 22syl 16 . . . . . 6  |-  ( ph  ->  (Scalar `  C )  e.  Grp )
2410lmodrng 15879 . . . . . . . 8  |-  ( C  e.  LMod  ->  (Scalar `  C )  e.  Ring )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  (Scalar `  C )  e.  Ring )
26 eqid 2381 . . . . . . . 8  |-  ( Base `  (Scalar `  C )
)  =  ( Base `  (Scalar `  C )
)
2726, 13rngidcl 15605 . . . . . . 7  |-  ( (Scalar `  C )  e.  Ring  -> 
( 1r `  (Scalar `  C ) )  e.  ( Base `  (Scalar `  C ) ) )
2825, 27syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  e.  ( Base `  (Scalar `  C ) ) )
2926, 12grpinvcl 14771 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  e.  (
Base `  (Scalar `  C
) ) )
311, 17, 19, 21, 2, 10, 26, 3lcdsbase 31767 . . . . 5  |-  ( ph  ->  ( Base `  (Scalar `  C ) )  =  ( Base `  R
) )
3230, 31eleqtrd 2457 . . . 4  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  e.  (
Base `  R )
)
331, 17, 19, 21, 2, 7, 11, 3, 32, 6lcdvscl 31772 . . 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 31765 . 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 2381 . . . . . . . . 9  |-  ( inv g `  R )  =  ( inv g `  R )
371, 17, 19, 36, 2, 10, 12, 3lcdneg 31777 . . . . . . . 8  |-  ( ph  ->  ( inv g `  (Scalar `  C ) )  =  ( inv g `  R ) )
38 eqid 2381 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  R
)
391, 17, 19, 38, 2, 10, 13, 3lcd1 31776 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  C ) )  =  ( 1r `  R
) )
4037, 39fveq12d 5668 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  (Scalar `  C )
) `  ( 1r `  (Scalar `  C )
) )  =  ( ( inv g `  R ) `  ( 1r `  R ) ) )
4140oveq1d 6029 . . . . . 6  |-  ( ph  ->  ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G )  =  ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  C ) G ) )
4241fveq1d 5664 . . . . 5  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) `  X )  =  ( ( ( ( inv g `  R ) `
 ( 1r `  R ) ) ( .s `  C ) G ) `  X
) )
43 eqid 2381 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
441, 17, 3dvhlmod 31277 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
4519lmodrng 15879 . . . . . . . . 9  |-  ( U  e.  LMod  ->  R  e. 
Ring )
4644, 45syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
47 rnggrp 15590 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4846, 47syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Grp )
4919, 21, 38lmod1cl 15898 . . . . . . . 8  |-  ( U  e.  LMod  ->  ( 1r
`  R )  e.  ( Base `  R
) )
5044, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
5121, 36grpinvcl 14771 . . . . . . 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
) )
531, 17, 18, 19, 21, 43, 2, 7, 11, 3, 52, 6, 34lcdvsval 31771 . . . . 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 31763 . . . . . 6  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  R ) )
5521, 43, 38, 36, 46, 54rngnegr 15625 . . . . 5  |-  ( ph  ->  ( ( G `  X ) ( .r
`  R ) ( ( inv g `  R ) `  ( 1r `  R ) ) )  =  ( ( inv g `  R
) `  ( G `  X ) ) )
5642, 53, 553eqtrd 2417 . . . 4  |-  ( ph  ->  ( ( ( ( inv g `  (Scalar `  C ) ) `  ( 1r `  (Scalar `  C ) ) ) ( .s `  C
) G ) `  X )  =  ( ( inv g `  R ) `  ( G `  X )
) )
5756oveq2d 6030 . . 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 31763 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  R ) )
59 lcdvsubval.s . . . . 5  |-  S  =  ( -g `  R
)
6021, 20, 36, 59grpsubval 14769 . . . 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 643 . . 3  |-  ( ph  ->  ( ( F `  X ) S ( G `  X ) )  =  ( ( F `  X ) ( +g  `  R
) ( ( inv g `  R ) `
 ( G `  X ) ) ) )
6257, 61eqtr4d 2416 . 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 2417 1  |-  ( ph  ->  ( ( F  .-  G ) `  X
)  =  ( ( F `  X ) S ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5388  (class class class)co 6014   Basecbs 13390   +g cplusg 13450   .rcmulr 13451  Scalarcsca 13453   .scvsca 13454   Grpcgrp 14606   inv gcminusg 14607   -gcsg 14609   Ringcrg 15581   1rcur 15583   LModclmod 15871   HLchlt 29517   LHypclh 30150   DVecHcdvh 31245  LCDualclcd 31753
This theorem is referenced by:  hdmapinvlem3  32090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-of 6238  df-1st 6282  df-2nd 6283  df-tpos 6409  df-undef 6473  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-n0 10148  df-z 10209  df-uz 10415  df-fz 10970  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-ress 13397  df-plusg 13463  df-mulr 13464  df-sca 13466  df-vsca 13467  df-0g 13648  df-mre 13732  df-mrc 13733  df-acs 13735  df-poset 14324  df-plt 14336  df-lub 14352  df-glb 14353  df-join 14354  df-meet 14355  df-p0 14389  df-p1 14390  df-lat 14396  df-clat 14458  df-mnd 14611  df-submnd 14660  df-grp 14733  df-minusg 14734  df-sbg 14735  df-subg 14862  df-cntz 15037  df-oppg 15063  df-lsm 15191  df-cmn 15335  df-abl 15336  df-mgp 15570  df-rng 15584  df-ur 15586  df-oppr 15649  df-dvdsr 15667  df-unit 15668  df-invr 15698  df-dvr 15709  df-drng 15758  df-lmod 15873  df-lss 15930  df-lsp 15969  df-lvec 16096  df-lsatoms 29143  df-lshyp 29144  df-lcv 29186  df-lfl 29225  df-lkr 29253  df-ldual 29291  df-oposet 29343  df-ol 29345  df-oml 29346  df-covers 29433  df-ats 29434  df-atl 29465  df-cvlat 29489  df-hlat 29518  df-llines 29664  df-lplanes 29665  df-lvols 29666  df-lines 29667  df-psubsp 29669  df-pmap 29670  df-padd 29962  df-lhyp 30154  df-laut 30155  df-ldil 30270  df-ltrn 30271  df-trl 30325  df-tgrp 30909  df-tendo 30921  df-edring 30923  df-dveca 31169  df-disoa 31196  df-dvech 31246  df-dib 31306  df-dic 31340  df-dih 31396  df-doch 31515  df-djh 31562  df-lcdual 31754
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