Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1eq Unicode version

Theorem hdmap1eq 32331
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 3319 . . . 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 32330 . . 3  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
1918eqeq1d 2438 . 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 32235 . . 3  |-  ( ph  ->  E! h  e.  D  ( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) )
23 nfv 1629 . . . 4  |-  F/ h ph
24 nfcvd 2567 . . . 4  |-  ( ph  -> 
F/_ h G )
25 nfvd 1630 . . . 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 3812 . . . . . . . 8  |-  ( h  =  G  ->  { h }  =  { G } )
2827fveq2d 5718 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { h } )  =  ( L `  { G } ) )
2928eqeq2d 2441 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) ) )
30 oveq2 6075 . . . . . . . . 9  |-  ( h  =  G  ->  ( F R h )  =  ( F R G ) )
3130sneqd 3814 . . . . . . . 8  |-  ( h  =  G  ->  { ( F R h ) }  =  { ( F R G ) } )
3231fveq2d 5718 . . . . . . 7  |-  ( h  =  G  ->  ( L `  { ( F R h ) } )  =  ( L `
 { ( F R G ) } ) )
3332eqeq2d 2441 . . . . . 6  |-  ( h  =  G  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( L `  { ( F R G ) } ) ) )
3429, 33anbi12d 692 . . . . 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 453 . . . 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 6556 . . 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 650 . 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 248 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2593   E!wreu 2694    \ cdif 3304   {csn 3801   <.cotp 3805   ` cfv 5440  (class class class)co 6067   iota_crio 6528   Basecbs 13452   0gc0g 13706   -gcsg 14671   LSpanclspn 16030   HLchlt 29879   LHypclh 30512   DVecHcdvh 31607  LCDualclcd 32115  mapdcmpd 32153  HDMap1chdma1 32321
This theorem is referenced by:  hdmap1l6lem1  32337  hdmap1l6lem2  32338  hdmap1l6a  32339  hdmap1neglem1N  32357  hdmapval3lemN  32369  hdmap10lem  32371  hdmap11lem1  32373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-ot 3811  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-tpos 6465  df-undef 6529  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-n0 10206  df-z 10267  df-uz 10473  df-fz 11028  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-sca 13528  df-vsca 13529  df-0g 13710  df-mre 13794  df-mrc 13795  df-acs 13797  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-mnd 14673  df-submnd 14722  df-grp 14795  df-minusg 14796  df-sbg 14797  df-subg 14924  df-cntz 15099  df-oppg 15125  df-lsm 15253  df-cmn 15397  df-abl 15398  df-mgp 15632  df-rng 15646  df-ur 15648  df-oppr 15711  df-dvdsr 15729  df-unit 15730  df-invr 15760  df-dvr 15771  df-drng 15820  df-lmod 15935  df-lss 15992  df-lsp 16031  df-lvec 16158  df-lsatoms 29505  df-lshyp 29506  df-lcv 29548  df-lfl 29587  df-lkr 29615  df-ldual 29653  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-lines 30029  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687  df-tgrp 31271  df-tendo 31283  df-edring 31285  df-dveca 31531  df-disoa 31558  df-dvech 31608  df-dib 31668  df-dic 31702  df-dih 31758  df-doch 31877  df-djh 31924  df-lcdual 32116  df-mapd 32154  df-hdmap1 32323
  Copyright terms: Public domain W3C validator