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Theorem mirbtwn 24703
Description: Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
Hypotheses
Ref Expression
mirval.p  |-  P  =  ( Base `  G
)
mirval.d  |-  .-  =  ( dist `  G )
mirval.i  |-  I  =  (Itv `  G )
mirval.l  |-  L  =  (LineG `  G )
mirval.s  |-  S  =  (pInvG `  G )
mirval.g  |-  ( ph  ->  G  e. TarskiG )
mirval.a  |-  ( ph  ->  A  e.  P )
mirfv.m  |-  M  =  ( S `  A
)
mirfv.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
mirbtwn  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )

Proof of Theorem mirbtwn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . . . 5  |-  P  =  ( Base `  G
)
2 mirval.d . . . . 5  |-  .-  =  ( dist `  G )
3 mirval.i . . . . 5  |-  I  =  (Itv `  G )
4 mirval.l . . . . 5  |-  L  =  (LineG `  G )
5 mirval.s . . . . 5  |-  S  =  (pInvG `  G )
6 mirval.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
7 mirval.a . . . . 5  |-  ( ph  ->  A  e.  P )
8 mirfv.m . . . . 5  |-  M  =  ( S `  A
)
9 mirfv.b . . . . 5  |-  ( ph  ->  B  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 24701 . . . 4  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
111, 2, 3, 6, 9, 7mirreu3 24699 . . . . 5  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
12 riotacl2 6265 . . . . 5  |-  ( E! z  e.  P  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) )  ->  ( iota_ z  e.  P  ( ( A 
.-  z )  =  ( A  .-  B
)  /\  A  e.  ( z I B ) ) )  e. 
{ z  e.  P  |  ( ( A 
.-  z )  =  ( A  .-  B
)  /\  A  e.  ( z I B ) ) } )
1311, 12syl 17 . . . 4  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  e.  {
z  e.  P  | 
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) } )
1410, 13eqeltrd 2529 . . 3  |-  ( ph  ->  ( M `  B
)  e.  { z  e.  P  |  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) ) } )
15 oveq2 6298 . . . . . 6  |-  ( z  =  ( M `  B )  ->  ( A  .-  z )  =  ( A  .-  ( M `  B )
) )
1615eqeq1d 2453 . . . . 5  |-  ( z  =  ( M `  B )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  B
) )  =  ( A  .-  B ) ) )
17 oveq1 6297 . . . . . 6  |-  ( z  =  ( M `  B )  ->  (
z I B )  =  ( ( M `
 B ) I B ) )
1817eleq2d 2514 . . . . 5  |-  ( z  =  ( M `  B )  ->  ( A  e.  ( z
I B )  <->  A  e.  ( ( M `  B ) I B ) ) )
1916, 18anbi12d 717 . . . 4  |-  ( z  =  ( M `  B )  ->  (
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) )  <->  ( ( A 
.-  ( M `  B ) )  =  ( A  .-  B
)  /\  A  e.  ( ( M `  B ) I B ) ) ) )
2019elrab 3196 . . 3  |-  ( ( M `  B )  e.  { z  e.  P  |  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) }  <->  ( ( M `  B )  e.  P  /\  (
( A  .-  ( M `  B )
)  =  ( A 
.-  B )  /\  A  e.  ( ( M `  B )
I B ) ) ) )
2114, 20sylib 200 . 2  |-  ( ph  ->  ( ( M `  B )  e.  P  /\  ( ( A  .-  ( M `  B ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  B )
I B ) ) ) )
2221simprrd 767 1  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   E!wreu 2739   {crab 2741   ` cfv 5582   iota_crio 6251  (class class class)co 6290   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485  pInvGcmir 24697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501  df-mir 24698
This theorem is referenced by:  mirmir  24707  mirinv  24711  miriso  24715  mirmir2  24719  mirln  24721  mirln2  24722  mirconn  24723  mirhl2  24726  mircgrextend  24727  mirtrcgr  24728  mirauto  24729  miduniq  24730  krippenlem  24735  ragflat  24749  ragcgr  24752  footex  24763  colperpexlem1  24772  colperpexlem3  24774  mideulem2  24776  opphllem  24777  opphllem1  24789  opphllem2  24790  opphllem4  24792  colhp  24812  midbtwn  24821  lmieu  24826  lmiisolem  24838
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