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Theorem hdmap1eq 31992
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.  }
) )
15 eldifi 3298 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1614, 15syl 15 . . . 4  |-  ( ph  ->  X  e.  V )
17 hdmap1eq.f . . . 4  |-  ( ph  ->  F  e.  D )
18 hdmap1eq.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18hdmap1val2 31991 . . 3  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
2019eqeq1d 2291 . 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 ) )
21 hdmap1eq.e . . . 4  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
22 hdmap1eq.mn . . . 4  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
231, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 18, 17, 21, 22mapdpg 31896 . . 3  |-  ( ph  ->  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )
24 nfv 1605 . . . 4  |-  F/ h ph
25 nfcvd 2420 . . . 4  |-  ( ph  -> 
F/_ h G )
26 nfvd 1606 . . . 4  |-  ( ph  ->  F/ h ( ( M `  ( N `
 { Y }
) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
27 hdmap1eq.g . . . 4  |-  ( ph  ->  G  e.  D )
28 sneq 3651 . . . . . . . 8  |-  ( h  =  G  ->  { h }  =  { G } )
2928fveq2d 5529 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { h } )  =  ( L `  { G } ) )
3029eqeq2d 2294 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) ) )
31 oveq2 5866 . . . . . . . . 9  |-  ( h  =  G  ->  ( F R h )  =  ( F R G ) )
3231sneqd 3653 . . . . . . . 8  |-  ( h  =  G  ->  { ( F R h ) }  =  { ( F R G ) } )
3332fveq2d 5529 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { ( F R h ) } )  =  ( L `
 { ( F R G ) } ) )
3433eqeq2d 2294 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) )
3530, 34anbi12d 691 . . . . 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 ) } ) ) ) )
3635adantl 452 . . . 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 ) } ) ) ) )
3724, 25, 26, 27, 36riota2df 6325 . . 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 ) )
3823, 37mpdan 649 . 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 ) )
3920, 38bitr4d 247 1  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E!wreu 2545    \ cdif 3149   {csn 3640   <.cotp 3644   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   0gc0g 13400   -gcsg 14365   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMap1chdma1 31982
This theorem is referenced by:  hdmap1l6lem1  31998  hdmap1l6lem2  31999  hdmap1l6a  32000  hdmap1neglem1N  32018  hdmapval3lemN  32030  hdmap10lem  32032  hdmap11lem1  32034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777  df-mapd 31815  df-hdmap1 31984
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