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Theorem hgmapval1 32012
Description: Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.)
Hypotheses
Ref Expression
hgmapval1.h  |-  H  =  ( LHyp `  K
)
hgmapval1.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapval1.r  |-  R  =  (Scalar `  U )
hgmapval1.i  |-  .1.  =  ( 1r `  R )
hgmapval1.g  |-  G  =  ( (HGMap `  K
) `  W )
hgmapval1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
hgmapval1  |-  ( ph  ->  ( G `  .1.  )  =  .1.  )

Proof of Theorem hgmapval1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hgmapval1.h . . 3  |-  H  =  ( LHyp `  K
)
2 hgmapval1.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 eqid 2388 . . 3  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2388 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapval1.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 31558 . 2  |-  ( ph  ->  E. x  e.  (
Base `  U )
x  =/=  ( 0g
`  U ) )
7 hgmapval1.r . . . . . . . . 9  |-  R  =  (Scalar `  U )
8 hgmapval1.i . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
9 eqid 2388 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2388 . . . . . . . . 9  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
11 eqid 2388 . . . . . . . . 9  |-  ( 1r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( 1r `  (Scalar `  ( (LCDual `  K
) `  W )
) )
121, 2, 7, 8, 9, 10, 11, 5lcd1 31725 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .1.  )
1312oveq1d 6036 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  (  .1.  ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) ) )
14133ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( 1r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  (  .1.  ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) ) )
151, 9, 5lcdlmod 31708 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
16153ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LMod )
17 eqid 2388 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
18 eqid 2388 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
1953ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 simp2 958 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  x  e.  (
Base `  U )
)
211, 2, 3, 9, 17, 18, 19, 20hdmapcl 31949 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  x )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
22 eqid 2388 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
2317, 10, 22, 11lmodvs1 15906 . . . . . . 7  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( (
(HDMap `  K ) `  W ) `  x
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )  ->  (
( 1r `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  ( ( (HDMap `  K
) `  W ) `  x ) )
2416, 21, 23syl2anc 643 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( 1r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  ( ( (HDMap `  K
) `  W ) `  x ) )
2514, 24eqtr3d 2422 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  (  .1.  ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  ( ( (HDMap `  K
) `  W ) `  x ) )
261, 2, 5dvhlmod 31226 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
27263ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
28 eqid 2388 . . . . . . . 8  |-  ( .s
`  U )  =  ( .s `  U
)
293, 7, 28, 8lmodvs1 15906 . . . . . . 7  |-  ( ( U  e.  LMod  /\  x  e.  ( Base `  U
) )  ->  (  .1.  ( .s `  U
) x )  =  x )
3027, 20, 29syl2anc 643 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  (  .1.  ( .s `  U ) x )  =  x )
3130fveq2d 5673 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  (  .1.  ( .s `  U ) x ) )  =  ( ( (HDMap `  K ) `  W
) `  x )
)
32 eqid 2388 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
33 hgmapval1.g . . . . . 6  |-  G  =  ( (HGMap `  K
) `  W )
347lmodrng 15886 . . . . . . . 8  |-  ( U  e.  LMod  ->  R  e. 
Ring )
3532, 8rngidcl 15612 . . . . . . . 8  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
3626, 34, 353syl 19 . . . . . . 7  |-  ( ph  ->  .1.  e.  ( Base `  R ) )
37363ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  .1.  e.  ( Base `  R ) )
381, 2, 3, 28, 7, 32, 9, 22, 18, 33, 19, 20, 37hgmapvs 32010 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  (  .1.  ( .s `  U ) x ) )  =  ( ( G `  .1.  ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) ) )
3925, 31, 383eqtr2rd 2427 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( G `
 .1.  ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  =  (  .1.  ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) ) )
40 eqid 2388 . . . . 5  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
41 eqid 2388 . . . . 5  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
421, 9, 5lcdlvec 31707 . . . . . 6  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
43423ad2ant1 978 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
441, 2, 7, 32, 33, 5, 36hgmapcl 32008 . . . . . . 7  |-  ( ph  ->  ( G `  .1.  )  e.  ( Base `  R ) )
451, 2, 7, 32, 9, 10, 40, 5lcdsbase 31716 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  (
Base `  R )
)
4644, 45eleqtrrd 2465 . . . . . 6  |-  ( ph  ->  ( G `  .1.  )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
47463ad2ant1 978 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( G `  .1.  )  e.  ( Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
4836, 45eleqtrrd 2465 . . . . . 6  |-  ( ph  ->  .1.  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
49483ad2ant1 978 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  .1.  e.  ( Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
50 simp3 959 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  x  =/=  ( 0g `  U ) )
511, 2, 3, 4, 9, 41, 18, 19, 20hdmapeq0 31963 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  x
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  x  =  ( 0g `  U ) ) )
5251necon3bid 2586 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  x
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  x  =/=  ( 0g `  U ) ) )
5350, 52mpbird 224 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  x )  =/=  ( 0g `  (
(LCDual `  K ) `  W ) ) )
5417, 22, 10, 40, 41, 43, 47, 49, 21, 53lvecvscan2 16112 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( ( G `  .1.  )
( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) )  =  (  .1.  ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) )  <->  ( G `  .1.  )  =  .1.  ) )
5539, 54mpbid 202 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( G `  .1.  )  =  .1.  )
5655rexlimdv3a 2776 . 2  |-  ( ph  ->  ( E. x  e.  ( Base `  U
) x  =/=  ( 0g `  U )  -> 
( G `  .1.  )  =  .1.  )
)
576, 56mpd 15 1  |-  ( ph  ->  ( G `  .1.  )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   Ringcrg 15588   1rcur 15590   LModclmod 15878   LVecclvec 16102   HLchlt 29466   LHypclh 30099   DVecHcdvh 31194  LCDualclcd 31702  HDMapchdma 31909  HGMapchg 32002
This theorem is referenced by:  hdmapglem5  32041
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-ot 3768  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-undef 6480  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-0g 13655  df-mre 13739  df-mrc 13740  df-acs 13742  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044  df-oppg 15070  df-lsm 15198  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lsatoms 29092  df-lshyp 29093  df-lcv 29135  df-lfl 29174  df-lkr 29202  df-ldual 29240  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-tgrp 30858  df-tendo 30870  df-edring 30872  df-dveca 31118  df-disoa 31145  df-dvech 31195  df-dib 31255  df-dic 31289  df-dih 31345  df-doch 31464  df-djh 31511  df-lcdual 31703  df-mapd 31741  df-hvmap 31873  df-hdmap1 31910  df-hdmap 31911  df-hgmap 32003
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