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Theorem hdmap1eq 31259
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 ) } ) ) ) )
Dummy variable  h is distinct from all other variables.

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 3299 . . . . 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 31258 . . 3  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
2019eqeq1d 2292 . 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 31163 . . 3  |-  ( ph  ->  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )
24 nfv 1606 . . . 4  |-  F/ h ph
25 nfcvd 2421 . . . 4  |-  ( ph  -> 
F/_ h G )
26 nfvd 1607 . . . 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 3652 . . . . . . . 8  |-  ( h  =  G  ->  { h }  =  { G } )
2928fveq2d 5489 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { h } )  =  ( L `  { G } ) )
3029eqeq2d 2295 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) ) )
31 oveq2 5827 . . . . . . . . 9  |-  ( h  =  G  ->  ( F R h )  =  ( F R G ) )
3231sneqd 3654 . . . . . . . 8  |-  ( h  =  G  ->  { ( F R h ) }  =  { ( F R G ) } )
3332fveq2d 5489 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { ( F R h ) } )  =  ( L `
 { ( F R G ) } ) )
3433eqeq2d 2295 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) )
3530, 34anbi12d 693 . . . . 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 6320 . . 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 651 . 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 1624    e. wcel 1685    =/= wne 2447   E!wreu 2546    \ cdif 3150   {csn 3641   <.cotp 3645   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   0gc0g 13394   -gcsg 14359   LSpanclspn 15722   HLchlt 28807   LHypclh 29440   DVecHcdvh 30535  LCDualclcd 31043  mapdcmpd 31081  HDMap1chdma1 31249
This theorem is referenced by:  hdmap1l6lem1  31265  hdmap1l6lem2  31266  hdmap1l6a  31267  hdmap1neglem1N  31285  hdmapval3lemN  31297  hdmap10lem  31299  hdmap11lem1  31301
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-ot 3651  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-undef 6291  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-cntz 14787  df-oppg 14813  df-lsm 14941  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-drng 15508  df-lmod 15623  df-lss 15684  df-lsp 15723  df-lvec 15850  df-lsatoms 28433  df-lshyp 28434  df-lcv 28476  df-lfl 28515  df-lkr 28543  df-ldual 28581  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tgrp 30199  df-tendo 30211  df-edring 30213  df-dveca 30459  df-disoa 30486  df-dvech 30536  df-dib 30596  df-dic 30630  df-dih 30686  df-doch 30805  df-djh 30852  df-lcdual 31044  df-mapd 31082  df-hdmap1 31251
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