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Theorem hgmapval0 32630
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 2435 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2435 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapval0.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 32177 . . 3  |-  ( ph  ->  E. x  e.  (
Base `  U )
x  =/=  ( 0g
`  U ) )
7 eqid 2435 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
8 eqid 2435 . . . . . . . . 9  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
9 eqid 2435 . . . . . . . . 9  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
105adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  x  e.  ( Base `  U )
)
121, 2, 3, 4, 7, 8, 9, 10, 11hdmapeq0 32582 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  x  =  ( 0g `  U ) ) )
1312biimpd 199 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  ->  x  =  ( 0g `  U ) ) )
1413necon3ad 2634 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( x  =/=  ( 0g `  U
)  ->  -.  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) )
15143impia 1150 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  -.  ( (
(HDMap `  K ) `  W ) `  x
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) ) )
161, 2, 5dvhlmod 31845 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  LMod )
17 hgmapval0.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  U )
18 eqid 2435 . . . . . . . . . . . . 13  |-  ( .s
`  U )  =  ( .s `  U
)
19 hgmapval0.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
203, 17, 18, 19, 4lmod0vs 15975 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  x  e.  ( Base `  U
) )  ->  (  .0.  ( .s `  U
) x )  =  ( 0g `  U
) )
2116, 20sylan 458 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  (  .0.  ( .s `  U ) x )  =  ( 0g `  U ) )
2221fveq2d 5724 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  (  .0.  ( .s `  U
) x ) )  =  ( ( (HDMap `  K ) `  W
) `  ( 0g `  U ) ) )
23 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
24 eqid 2435 . . . . . . . . . . 11  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
25 hgmapval0.g . . . . . . . . . . 11  |-  G  =  ( (HGMap `  K
) `  W )
2617, 23, 19lmod0cl 15968 . . . . . . . . . . . . 13  |-  ( U  e.  LMod  ->  .0.  e.  ( Base `  R )
)
2716, 26syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
2827adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  .0.  e.  ( Base `  R )
)
291, 2, 3, 18, 17, 23, 7, 24, 9, 25, 10, 11, 28hgmapvs 32629 . . . . . . . . . 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 32571 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (HDMap `  K ) `  W
) `  ( 0g `  U ) )  =  ( 0g `  (
(LCDual `  K ) `  W ) ) )
3130adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  ( 0g `  U ) )  =  ( 0g `  ( (LCDual `  K ) `  W ) ) )
3222, 29, 313eqtr3d 2475 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( ( G `  .0.  ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) )
33 eqid 2435 . . . . . . . . . 10  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
34 eqid 2435 . . . . . . . . . 10  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
35 eqid 2435 . . . . . . . . . 10  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
36 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )
371, 7, 5lcdlvec 32326 . . . . . . . . . . 11  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
3837adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
391, 2, 10dvhlmod 31845 . . . . . . . . . . . 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 32628 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( G `  .0.  )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
421, 2, 3, 7, 33, 9, 10, 11hdmapcl 32568 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  x
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
4333, 24, 34, 35, 36, 8, 38, 41, 42lvecvs0or 16172 . . . . . . . . 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 202 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  \/  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) )
4544orcomd 378 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  \/  ( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ) )
4645ord 367 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( -.  ( ( (HDMap `  K ) `  W
) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W ) )  -> 
( G `  .0.  )  =  ( 0g `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ) )
47463adant3 977 . . . . 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 2824 . . 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 32343 . 2  |-  ( ph  ->  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .0.  )
5250, 51eqtrd 2467 1  |-  ( ph  ->  ( G `  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LModclmod 15942   LVecclvec 16166   HLchlt 30085   LHypclh 30718   DVecHcdvh 31813  LCDualclcd 32321  HDMapchdma 32528  HGMapchg 32621
This theorem is referenced by:  hgmapeq0  32642  hgmapvv  32664
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lshyp 29712  df-lcv 29754  df-lfl 29793  df-lkr 29821  df-ldual 29859  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tgrp 31477  df-tendo 31489  df-edring 31491  df-dveca 31737  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083  df-djh 32130  df-lcdual 32322  df-mapd 32360  df-hvmap 32492  df-hdmap1 32529  df-hdmap 32530  df-hgmap 32622
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