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Theorem mirbtwn 23752
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 . . . . . 6  |-  P  =  ( Base `  G
)
2 mirval.d . . . . . 6  |-  .-  =  ( dist `  G )
3 mirval.i . . . . . 6  |-  I  =  (Itv `  G )
4 mirval.l . . . . . 6  |-  L  =  (LineG `  G )
5 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
6 mirval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
7 mirval.a . . . . . 6  |-  ( ph  ->  A  e.  P )
8 mirfv.m . . . . . 6  |-  M  =  ( S `  A
)
9 mirfv.b . . . . . 6  |-  ( ph  ->  B  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 23750 . . . . 5  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
111, 2, 3, 6, 9, 7mirreu3 23748 . . . . . 6  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
12 riotacl2 6257 . . . . . 6  |-  ( 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 16 . . . . 5  |-  ( 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 2555 . . . 4  |-  ( ph  ->  ( M `  B
)  e.  { z  e.  P  |  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) ) } )
15 oveq2 6290 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  ( A  .-  z )  =  ( A  .-  ( M `  B )
) )
1615eqeq1d 2469 . . . . . 6  |-  ( z  =  ( M `  B )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  B
) )  =  ( A  .-  B ) ) )
17 oveq1 6289 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  (
z I B )  =  ( ( M `
 B ) I B ) )
1817eleq2d 2537 . . . . . 6  |-  ( z  =  ( M `  B )  ->  ( A  e.  ( z
I B )  <->  A  e.  ( ( M `  B ) I B ) ) )
1916, 18anbi12d 710 . . . . 5  |-  ( z  =  ( M `  B )  ->  (
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) )  <->  ( ( A 
.-  ( M `  B ) )  =  ( A  .-  B
)  /\  A  e.  ( ( M `  B ) I B ) ) ) )
2019elrab 3261 . . . 4  |-  ( ( 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 196 . . 3  |-  ( ph  ->  ( ( M `  B )  e.  P  /\  ( ( A  .-  ( M `  B ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  B )
I B ) ) ) )
2221simprd 463 . 2  |-  ( ph  ->  ( ( A  .-  ( M `  B ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  B )
I B ) ) )
2322simprd 463 1  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2816   {crab 2818   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486   distcds 14560  TarskiGcstrkg 23553  Itvcitv 23560  LineGclng 23561  pInvGcmir 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-trkgc 23572  df-trkgb 23573  df-trkgcb 23574  df-trkg 23578  df-mir 23747
This theorem is referenced by:  mirmir  23756  mirinv  23760  miriso  23763  mirmir2  23767  mirauto  23769  miduniq  23770  krippenlem  23775  ragflat  23789  ragcgr  23792  footex  23803  colperpexlem1  23809  colperpexlem3  23811  mideulem  23813  midbtwn  23822  lmieu  23827  lmiisolem  23838
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