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Theorem mircgr 23744
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 23743 . . . . 5  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
111, 2, 3, 6, 9, 7mirreu3 23741 . . . . . 6  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
12 riotacl2 6250 . . . . . 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 2548 . . . 4  |-  ( ph  ->  ( M `  B
)  e.  { z  e.  P  |  ( ( A  .-  z
)  =  ( A 
.-  B )  /\  A  e.  ( z
I B ) ) } )
15 oveq2 6283 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  ( A  .-  z )  =  ( A  .-  ( M `  B )
) )
1615eqeq1d 2462 . . . . . 6  |-  ( z  =  ( M `  B )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  B
) )  =  ( A  .-  B ) ) )
17 oveq1 6282 . . . . . . 7  |-  ( z  =  ( M `  B )  ->  (
z I B )  =  ( ( M `
 B ) I B ) )
1817eleq2d 2530 . . . . . 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 3254 . . . 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 1374    e. wcel 1762   E!wreu 2809   {crab 2811   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  pInvGcmir 23739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571  df-mir 23740
This theorem is referenced by:  mirmir  23749  miriso  23756  mirmir2  23760  mirauto  23762  miduniq  23763  krippenlem  23768  ragcol  23777  ragflat  23782  ragcgr  23785  footex  23796  colperpexlem1  23802  colperpexlem3  23804  mideulem  23806  midcgr  23816  lmiisolem  23831
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