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Theorem mircgr 23189
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
mircgr  |-  ( ph  ->  ( A  .-  ( M `  B )
)  =  ( A 
.-  B ) )

Proof of Theorem mircgr
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 23188 . . . . 5  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
111, 2, 3, 6, 9, 7mirreu3 23186 . . . . . 6  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
12 riotacl2 6167 . . . . . 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 2539 . . . 4  |-  ( ph  ->  ( M `  B
)  e.  { z  e.  P  |  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) ) } )
15 oveq2 6200 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  ( A  .-  z )  =  ( A  .-  ( M `  B )
) )
1615eqeq1d 2453 . . . . . 6  |-  ( z  =  ( M `  B )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  B
) )  =  ( A  .-  B ) ) )
17 oveq1 6199 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  (
z I B )  =  ( ( M `
 B ) I B ) )
1817eleq2d 2521 . . . . . 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 3216 . . . 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 ) ) )
2322simpld 459 1  |-  ( ph  ->  ( A  .-  ( M `  B )
)  =  ( A 
.-  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E!wreu 2797   {crab 2799   ` cfv 5518   iota_crio 6152  (class class class)co 6192   Basecbs 14278   distcds 14351  TarskiGcstrkg 23007  Itvcitv 23014  LineGclng 23015  pInvGcmir 23183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-trkgc 23026  df-trkgb 23027  df-trkgcb 23028  df-trkg 23032  df-mir 23184
This theorem is referenced by:  mirmir  23194  miriso  23201  mirmir2  23205  mirauto  23206  miduniq  23207  krippenlem  23212  ragcol  23221  ragflat  23226  ragcgr  23229  footex  23239  colperplem1  23242  colperplem3  23244  mideulem  23246
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