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Theorem hdmap1eq 35447
Description: The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 16-May-2015.)
Hypotheses
Ref Expression
hdmap1val2.h  |-  H  =  ( LHyp `  K
)
hdmap1val2.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1val2.v  |-  V  =  ( Base `  U
)
hdmap1val2.s  |-  .-  =  ( -g `  U )
hdmap1val2.o  |-  .0.  =  ( 0g `  U )
hdmap1val2.n  |-  N  =  ( LSpan `  U )
hdmap1val2.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1val2.d  |-  D  =  ( Base `  C
)
hdmap1val2.r  |-  R  =  ( -g `  C
)
hdmap1val2.l  |-  L  =  ( LSpan `  C )
hdmap1val2.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1val2.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1val2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1eq.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1eq.f  |-  ( ph  ->  F  e.  D )
hdmap1eq.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap1eq.g  |-  ( ph  ->  G  e.  D )
hdmap1eq.e  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
hdmap1eq.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
Assertion
Ref Expression
hdmap1eq  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )

Proof of Theorem hdmap1eq
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 hdmap1val2.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmap1val2.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1val2.v . . . 4  |-  V  =  ( Base `  U
)
4 hdmap1val2.s . . . 4  |-  .-  =  ( -g `  U )
5 hdmap1val2.o . . . 4  |-  .0.  =  ( 0g `  U )
6 hdmap1val2.n . . . 4  |-  N  =  ( LSpan `  U )
7 hdmap1val2.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap1val2.d . . . 4  |-  D  =  ( Base `  C
)
9 hdmap1val2.r . . . 4  |-  R  =  ( -g `  C
)
10 hdmap1val2.l . . . 4  |-  L  =  ( LSpan `  C )
11 hdmap1val2.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
12 hdmap1val2.i . . . 4  |-  I  =  ( (HDMap1 `  K
) `  W )
13 hdmap1val2.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
14 hdmap1eq.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1514eldifad 3340 . . . 4  |-  ( ph  ->  X  e.  V )
16 hdmap1eq.f . . . 4  |-  ( ph  ->  F  e.  D )
17 hdmap1eq.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17hdmap1val2 35446 . . 3  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D  ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
1918eqeq1d 2451 . 2  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( iota_ h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )  =  G ) )
20 hdmap1eq.e . . . 4  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
21 hdmap1eq.mn . . . 4  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
221, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 17, 16, 20, 21mapdpg 35351 . . 3  |-  ( ph  ->  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )
23 nfv 1673 . . . 4  |-  F/ h ph
24 nfcvd 2580 . . . 4  |-  ( ph  -> 
F/_ h G )
25 nfvd 1674 . . . 4  |-  ( ph  ->  F/ h ( ( M `  ( N `
 { Y }
) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
26 hdmap1eq.g . . . 4  |-  ( ph  ->  G  e.  D )
27 sneq 3887 . . . . . . . 8  |-  ( h  =  G  ->  { h }  =  { G } )
2827fveq2d 5695 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { h } )  =  ( L `  { G } ) )
2928eqeq2d 2454 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) ) )
30 oveq2 6099 . . . . . . . . 9  |-  ( h  =  G  ->  ( F R h )  =  ( F R G ) )
3130sneqd 3889 . . . . . . . 8  |-  ( h  =  G  ->  { ( F R h ) }  =  { ( F R G ) } )
3231fveq2d 5695 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { ( F R h ) } )  =  ( L `
 { ( F R G ) } ) )
3332eqeq2d 2454 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) )
3429, 33anbi12d 710 . . . . 5  |-  ( h  =  G  ->  (
( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) )  <->  ( ( M `  ( N `  { Y } ) )  =  ( L `
 { G }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) ) )
3534adantl 466 . . . 4  |-  ( (
ph  /\  h  =  G )  ->  (
( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) )  <->  ( ( M `  ( N `  { Y } ) )  =  ( L `
 { G }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) ) )
3623, 24, 25, 26, 35riota2df 6073 . . 3  |-  ( (
ph  /\  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `
 { h }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R h ) } ) ) )  ->  (
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) )  <->  ( iota_ h  e.  D  ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) )  =  G ) )
3722, 36mpdan 668 . 2  |-  ( ph  ->  ( ( ( M `
 ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R G ) } ) )  <->  ( iota_ h  e.  D  ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) )  =  G ) )
3819, 37bitr4d 256 1  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E!wreu 2717    \ cdif 3325   {csn 3877   <.cotp 3885   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   0gc0g 14378   -gcsg 15413   LSpanclspn 17052   HLchlt 32995   LHypclh 33628   DVecHcdvh 34723  LCDualclcd 35231  mapdcmpd 35269  HDMap1chdma1 35437
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-riotaBAD 32604
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-undef 6792  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-0g 14380  df-mre 14524  df-mrc 14525  df-acs 14527  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-oppg 15861  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lsatoms 32621  df-lshyp 32622  df-lcv 32664  df-lfl 32703  df-lkr 32731  df-ldual 32769  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803  df-tgrp 34387  df-tendo 34399  df-edring 34401  df-dveca 34647  df-disoa 34674  df-dvech 34724  df-dib 34784  df-dic 34818  df-dih 34874  df-doch 34993  df-djh 35040  df-lcdual 35232  df-mapd 35270  df-hdmap1 35439
This theorem is referenced by:  hdmap1l6lem1  35453  hdmap1l6lem2  35454  hdmap1l6a  35455  hdmap1neglem1N  35473  hdmapval3lemN  35485  hdmap10lem  35487  hdmap11lem1  35489
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