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Theorem hdmap1eq 31159
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
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 3273 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1614, 15syl 17 . . . 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 31158 . . 3  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
2019eqeq1d 2266 . 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 31063 . . 3  |-  ( ph  ->  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )
24 nfv 1629 . . . 4  |-  F/ h ph
25 nfcvd 2395 . . . 4  |-  ( ph  -> 
F/_ h G )
26 nfvd 1631 . . . 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 3625 . . . . . . . 8  |-  ( h  =  G  ->  { h }  =  { G } )
2928fveq2d 5462 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { h } )  =  ( L `  { G } ) )
3029eqeq2d 2269 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) ) )
31 oveq2 5800 . . . . . . . . 9  |-  ( h  =  G  ->  ( F R h )  =  ( F R G ) )
3231sneqd 3627 . . . . . . . 8  |-  ( h  =  G  ->  { ( F R h ) }  =  { ( F R G ) } )
3332fveq2d 5462 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { ( F R h ) } )  =  ( L `
 { ( F R G ) } ) )
3433eqeq2d 2269 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) )
3530, 34anbi12d 694 . . . . 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 454 . . . 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 6293 . . 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 652 . 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 249 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E!wreu 2520    \ cdif 3124   {csn 3614   <.cotp 3618   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13110   0gc0g 13362   -gcsg 14327   LSpanclspn 15690   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  mapdcmpd 30981  HDMap1chdma1 31149
This theorem is referenced by:  hdmap1l6lem1  31165  hdmap1l6lem2  31166  hdmap1l6a  31167  hdmap1neglem1N  31185  hdmapval3lemN  31197  hdmap10lem  31199  hdmap11lem1  31201
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982  df-hdmap1 31151
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