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Theorem hgmapval1 37325
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 2441 . . 3  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2441 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapval1.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 36871 . 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 2441 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2441 . . . . . . . . 9  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
11 eqid 2441 . . . . . . . . 9  |-  ( 1r
`  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( 1r `  (Scalar `  ( (LCDual `  K
) `  W )
) )
121, 2, 7, 8, 9, 10, 11, 5lcd1 37038 . . . . . . . 8  |-  ( ph  ->  ( 1r `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .1.  )
1312oveq1d 6292 . . . . . . 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 1016 . . . . . 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 37021 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
16153ad2ant1 1016 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LMod )
17 eqid 2441 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
18 eqid 2441 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
1953ad2ant1 1016 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 simp2 996 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  x  e.  (
Base `  U )
)
211, 2, 3, 9, 17, 18, 19, 20hdmapcl 37262 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  x )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
22 eqid 2441 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
2317, 10, 22, 11lmodvs1 17408 . . . . . . 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 661 . . . . . 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 2484 . . . . 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 36539 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
27263ad2ant1 1016 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
28 eqid 2441 . . . . . . . 8  |-  ( .s
`  U )  =  ( .s `  U
)
293, 7, 28, 8lmodvs1 17408 . . . . . . 7  |-  ( ( U  e.  LMod  /\  x  e.  ( Base `  U
) )  ->  (  .1.  ( .s `  U
) x )  =  x )
3027, 20, 29syl2anc 661 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  (  .1.  ( .s `  U ) x )  =  x )
3130fveq2d 5856 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  (  .1.  ( .s `  U ) x ) )  =  ( ( (HDMap `  K ) `  W
) `  x )
)
32 eqid 2441 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
33 hgmapval1.g . . . . . 6  |-  G  =  ( (HGMap `  K
) `  W )
347lmodring 17388 . . . . . . . 8  |-  ( U  e.  LMod  ->  R  e. 
Ring )
3532, 8ringidcl 17087 . . . . . . . 8  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
3626, 34, 353syl 20 . . . . . . 7  |-  ( ph  ->  .1.  e.  ( Base `  R ) )
37363ad2ant1 1016 . . . . . 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 37323 . . . . 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 2489 . . . 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 2441 . . . . 5  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
41 eqid 2441 . . . . 5  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
421, 9, 5lcdlvec 37020 . . . . . 6  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
43423ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
441, 2, 7, 32, 33, 5, 36hgmapcl 37321 . . . . . . 7  |-  ( ph  ->  ( G `  .1.  )  e.  ( Base `  R ) )
451, 2, 7, 32, 9, 10, 40, 5lcdsbase 37029 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  (
Base `  R )
)
4644, 45eleqtrrd 2532 . . . . . 6  |-  ( ph  ->  ( G `  .1.  )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
47463ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( G `  .1.  )  e.  ( Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
4836, 45eleqtrrd 2532 . . . . . 6  |-  ( ph  ->  .1.  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
49483ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  .1.  e.  ( Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
50 simp3 997 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  x  =/=  ( 0g `  U ) )
511, 2, 3, 4, 9, 41, 18, 19, 20hdmapeq0 37276 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  x
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  x  =  ( 0g `  U ) ) )
5251necon3bid 2699 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  x
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  x  =/=  ( 0g `  U ) ) )
5350, 52mpbird 232 . . . . 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 17626 . . . 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 210 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  U )  /\  x  =/=  ( 0g `  U ) )  ->  ( G `  .1.  )  =  .1.  )
5655rexlimdv3a 2935 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792   ` cfv 5574  (class class class)co 6277   Basecbs 14504  Scalarcsca 14572   .scvsca 14573   0gc0g 14709   1rcur 17021   Ringcrg 17066   LModclmod 17380   LVecclvec 17616   HLchlt 34777   LHypclh 35410   DVecHcdvh 36507  LCDualclcd 37015  HDMapchdma 37222  HGMapchg 37315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-riotaBAD 34386
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-ot 4019  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-undef 7000  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-0g 14711  df-mre 14855  df-mrc 14856  df-acs 14858  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-cntz 16224  df-oppg 16250  df-lsm 16525  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-dvr 17200  df-drng 17266  df-lmod 17382  df-lss 17447  df-lsp 17486  df-lvec 17617  df-lsatoms 34403  df-lshyp 34404  df-lcv 34446  df-lfl 34485  df-lkr 34513  df-ldual 34551  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925  df-lvols 34926  df-lines 34927  df-psubsp 34929  df-pmap 34930  df-padd 35222  df-lhyp 35414  df-laut 35415  df-ldil 35530  df-ltrn 35531  df-trl 35586  df-tgrp 36171  df-tendo 36183  df-edring 36185  df-dveca 36431  df-disoa 36458  df-dvech 36508  df-dib 36568  df-dic 36602  df-dih 36658  df-doch 36777  df-djh 36824  df-lcdual 37016  df-mapd 37054  df-hvmap 37186  df-hdmap1 37223  df-hdmap 37224  df-hgmap 37316
This theorem is referenced by:  hdmapglem5  37354
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