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Theorem mirbtwn 24165
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 24163 . . . 4  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
111, 2, 3, 6, 9, 7mirreu3 24161 . . . . 5  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
12 riotacl2 6271 . . . . 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 16 . . . 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 2545 . . 3  |-  ( ph  ->  ( M `  B
)  e.  { z  e.  P  |  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) ) } )
15 oveq2 6304 . . . . . 6  |-  ( z  =  ( M `  B )  ->  ( A  .-  z )  =  ( A  .-  ( M `  B )
) )
1615eqeq1d 2459 . . . . 5  |-  ( z  =  ( M `  B )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  B
) )  =  ( A  .-  B ) ) )
17 oveq1 6303 . . . . . 6  |-  ( z  =  ( M `  B )  ->  (
z I B )  =  ( ( M `
 B ) I B ) )
1817eleq2d 2527 . . . . 5  |-  ( z  =  ( M `  B )  ->  ( A  e.  ( z
I B )  <->  A  e.  ( ( M `  B ) I B ) ) )
1916, 18anbi12d 710 . . . 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 3257 . . 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 196 . 2  |-  ( ph  ->  ( ( M `  B )  e.  P  /\  ( ( A  .-  ( M `  B ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  B )
I B ) ) ) )
2221simprrd 758 1  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E!wreu 2809   {crab 2811   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959  pInvGcmir 24159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-mir 24160
This theorem is referenced by:  mirmir  24169  mirinv  24173  miriso  24176  mirmir2  24180  mirln  24182  mirln2  24183  mirconn  24184  mirauto  24187  miduniq  24188  krippenlem  24193  ragflat  24207  ragcgr  24210  footex  24221  colperpexlem1  24230  colperpexlem3  24232  mideulem2  24234  opphllem  24235  opphllem1  24245  opphllem2  24246  opphllem4  24248  midbtwn  24271  lmieu  24276  lmiisolem  24287
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