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Theorem hgmapval0 35543
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 35090 . . 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 35495 . . . . . . . 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 2647 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( x  =/=  ( 0g `  U
)  ->  -.  (
( (HDMap `  K
) `  W ) `  x )  =  ( 0g `  ( (LCDual `  K ) `  W
) ) ) )
15143impia 1184 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  -.  ( (
(HDMap `  K ) `  W ) `  x
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) ) )
161, 2, 5dvhlmod 34758 . . . . . . . . . . . 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 16984 . . . . . . . . . . . 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 5698 . . . . . . . . . 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 16977 . . . . . . . . . . . . 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 35542 . . . . . . . . . 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 35484 . . . . . . . . . . 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 2483 . . . . . . . . 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 35239 . . . . . . . . . . 11  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
3837adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
391, 2, 10dvhlmod 34758 . . . . . . . . . . . 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 35541 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( G `  .0.  )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
421, 2, 3, 7, 33, 9, 10, 11hdmapcl 35481 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  U )
)  ->  ( (
(HDMap `  K ) `  W ) `  x
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
4333, 24, 34, 35, 36, 8, 38, 41, 42lvecvs0or 17192 . . . . . . . . 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 1008 . . . . 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 2846 . . 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 35256 . 2  |-  ( ph  ->  ( 0g `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .0.  )
5250, 51eqtrd 2475 1  |-  ( ph  ->  ( G `  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2609   E.wrex 2719   ` cfv 5421  (class class class)co 6094   Basecbs 14177  Scalarcsca 14244   .scvsca 14245   0gc0g 14381   LModclmod 16951   LVecclvec 17186   HLchlt 32998   LHypclh 33631   DVecHcdvh 34726  LCDualclcd 35234  HDMapchdma 35441  HGMapchg 35534
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-riotaBAD 32607
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-ot 3889  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-tpos 6748  df-undef 6795  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-n0 10583  df-z 10650  df-uz 10865  df-fz 11441  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-sca 14257  df-vsca 14258  df-0g 14383  df-mre 14527  df-mrc 14528  df-acs 14530  df-poset 15119  df-plt 15131  df-lub 15147  df-glb 15148  df-join 15149  df-meet 15150  df-p0 15212  df-p1 15213  df-lat 15219  df-clat 15281  df-mnd 15418  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-subg 15681  df-cntz 15838  df-oppg 15864  df-lsm 16138  df-cmn 16282  df-abl 16283  df-mgp 16595  df-ur 16607  df-rng 16650  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-dvr 16778  df-drng 16837  df-lmod 16953  df-lss 17017  df-lsp 17056  df-lvec 17187  df-lsatoms 32624  df-lshyp 32625  df-lcv 32667  df-lfl 32706  df-lkr 32734  df-ldual 32772  df-oposet 32824  df-ol 32826  df-oml 32827  df-covers 32914  df-ats 32915  df-atl 32946  df-cvlat 32970  df-hlat 32999  df-llines 33145  df-lplanes 33146  df-lvols 33147  df-lines 33148  df-psubsp 33150  df-pmap 33151  df-padd 33443  df-lhyp 33635  df-laut 33636  df-ldil 33751  df-ltrn 33752  df-trl 33806  df-tgrp 34390  df-tendo 34402  df-edring 34404  df-dveca 34650  df-disoa 34677  df-dvech 34727  df-dib 34787  df-dic 34821  df-dih 34877  df-doch 34996  df-djh 35043  df-lcdual 35235  df-mapd 35273  df-hvmap 35405  df-hdmap1 35442  df-hdmap 35443  df-hgmap 35535
This theorem is referenced by:  hgmapeq0  35555  hgmapvv  35577
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