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Theorem hgmapval0 37362
Description: Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.)
Hypotheses
Ref Expression
hgmapval0.h  |-  H  =  ( LHyp `  K
)
hgmapval0.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapval0.r  |-  R  =  (Scalar `  U )
hgmapval0.o  |-  .0.  =  ( 0g `  R )
hgmapval0.g  |-  G  =  ( (HGMap `  K
) `  W )
hgmapval0.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
hgmapval0  |-  ( ph  ->  ( G `  .0.  )  =  .0.  )

Proof of Theorem hgmapval0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hgmapval0.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hgmapval0.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 eqid 2443 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2443 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapval0.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 36909 . . 3  |-  ( ph  ->  E. x  e.  (
Base `  U )
x  =/=  ( 0g
`  U ) )
7 eqid 2443 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
8 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
9 eqid 2443 . . . . . . . . 9  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
105adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  x  e.  ( Base `  U )
)
121, 2, 3, 4, 7, 8, 9, 10, 11hdmapeq0 37314 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  x  =  ( 0g `  U ) ) )
1312biimpd 207 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  ->  x  =  ( 0g `  U ) ) )
1413necon3ad 2653 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( x  =/=  ( 0g `  U
)  ->  -.  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) )
15143impia 1194 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  -.  ( (
(HDMap `  K ) `  W ) `  x
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) ) )
161, 2, 5dvhlmod 36577 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  LMod )
17 hgmapval0.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  U )
18 eqid 2443 . . . . . . . . . . . . 13  |-  ( .s
`  U )  =  ( .s `  U
)
19 hgmapval0.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
203, 17, 18, 19, 4lmod0vs 17523 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  x  e.  ( Base `  U
) )  ->  (  .0.  ( .s `  U
) x )  =  ( 0g `  U
) )
2116, 20sylan 471 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  (  .0.  ( .s `  U ) x )  =  ( 0g `  U ) )
2221fveq2d 5860 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  (  .0.  ( .s `  U
) x ) )  =  ( ( (HDMap `  K ) `  W
) `  ( 0g `  U ) ) )
23 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
24 eqid 2443 . . . . . . . . . . 11  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
25 hgmapval0.g . . . . . . . . . . 11  |-  G  =  ( (HGMap `  K
) `  W )
2617, 23, 19lmod0cl 17516 . . . . . . . . . . . . 13  |-  ( U  e.  LMod  ->  .0.  e.  ( Base `  R )
)
2716, 26syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2827adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  .0.  e.  ( Base `  R )
)
291, 2, 3, 18, 17, 23, 7, 24, 9, 25, 10, 11, 28hgmapvs 37361 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  (  .0.  ( .s `  U
) x ) )  =  ( ( G `
 .0.  ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) )
301, 2, 4, 7, 8, 9, 5hdmapval0 37303 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (HDMap `  K ) `  W
) `  ( 0g `  U ) )  =  ( 0g `  (
(LCDual `  K ) `  W ) ) )
3130adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  ( 0g `  U ) )  =  ( 0g `  ( (LCDual `  K ) `  W ) ) )
3222, 29, 313eqtr3d 2492 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( ( G `  .0.  ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) )
33 eqid 2443 . . . . . . . . . 10  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
34 eqid 2443 . . . . . . . . . 10  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
35 eqid 2443 . . . . . . . . . 10  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
36 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )
371, 7, 5lcdlvec 37058 . . . . . . . . . . 11  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
3837adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
391, 2, 10dvhlmod 36577 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  U  e.  LMod )
4039, 26syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  .0.  e.  ( Base `  R )
)
411, 2, 17, 23, 7, 34, 35, 25, 10, 40hgmapdcl 37360 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( G `  .0.  )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
421, 2, 3, 7, 33, 9, 10, 11hdmapcl 37300 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  x
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
4333, 24, 34, 35, 36, 8, 38, 41, 42lvecvs0or 17732 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( G `  .0.  ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) )  =  ( 0g `  (
(LCDual `  K ) `  W ) )  <->  ( ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  \/  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) ) )
4432, 43mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  \/  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) )
4544orcomd 388 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  \/  ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ) )
4645ord 377 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( -.  ( ( (HDMap `  K ) `  W
) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W ) )  -> 
( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ) )
47463adant3 1017 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( -.  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  ->  ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ) )
4815, 47mpd 15 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( G `  .0.  )  =  ( 0g `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
4948rexlimdv3a 2937 . . 3  |-  ( ph  ->  ( E. x  e.  ( Base `  U
) x  =/=  ( 0g `  U )  -> 
( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ) )
506, 49mpd 15 . 2  |-  ( ph  ->  ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
511, 2, 17, 19, 7, 34, 36, 5lcd0 37075 . 2  |-  ( ph  ->  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .0.  )
5250, 51eqtrd 2484 1  |-  ( ph  ->  ( G `  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   ` cfv 5578  (class class class)co 6281   Basecbs 14613  Scalarcsca 14681   .scvsca 14682   0gc0g 14818   LModclmod 17490   LVecclvec 17726   HLchlt 34815   LHypclh 35448   DVecHcdvh 36545  LCDualclcd 37053  HDMapchdma 37260  HGMapchg 37353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-undef 7004  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-n0 10803  df-z 10872  df-uz 11092  df-fz 11683  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-0g 14820  df-mre 14964  df-mrc 14965  df-acs 14967  df-preset 15535  df-poset 15553  df-plt 15566  df-lub 15582  df-glb 15583  df-join 15584  df-meet 15585  df-p0 15647  df-p1 15648  df-lat 15654  df-clat 15716  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-subg 16176  df-cntz 16333  df-oppg 16359  df-lsm 16634  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-oppr 17250  df-dvdsr 17268  df-unit 17269  df-invr 17299  df-dvr 17310  df-drng 17376  df-lmod 17492  df-lss 17557  df-lsp 17596  df-lvec 17727  df-lsatoms 34441  df-lshyp 34442  df-lcv 34484  df-lfl 34523  df-lkr 34551  df-ldual 34589  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964  df-lines 34965  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624  df-tgrp 36209  df-tendo 36221  df-edring 36223  df-dveca 36469  df-disoa 36496  df-dvech 36546  df-dib 36606  df-dic 36640  df-dih 36696  df-doch 36815  df-djh 36862  df-lcdual 37054  df-mapd 37092  df-hvmap 37224  df-hdmap1 37261  df-hdmap 37262  df-hgmap 37354
This theorem is referenced by:  hgmapeq0  37374  hgmapvv  37396
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