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Theorem ismir 23905
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 )
ismir.1  |-  ( ph  ->  C  e.  P )
ismir.2  |-  ( ph  ->  ( A  .-  C
)  =  ( A 
.-  B ) )
ismir.3  |-  ( ph  ->  A  e.  ( C I B ) )
Assertion
Ref Expression
ismir  |-  ( ph  ->  C  =  ( M `
 B ) )

Proof of Theorem ismir
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . 3  |-  P  =  ( Base `  G
)
2 mirval.d . . 3  |-  .-  =  ( dist `  G )
3 mirval.i . . 3  |-  I  =  (Itv `  G )
4 mirval.l . . 3  |-  L  =  (LineG `  G )
5 mirval.s . . 3  |-  S  =  (pInvG `  G )
6 mirval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
7 mirval.a . . 3  |-  ( ph  ->  A  e.  P )
8 mirfv.m . . 3  |-  M  =  ( S `  A
)
9 mirfv.b . . 3  |-  ( ph  ->  B  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 23902 . 2  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
11 ismir.2 . . 3  |-  ( ph  ->  ( A  .-  C
)  =  ( A 
.-  B ) )
12 ismir.3 . . 3  |-  ( ph  ->  A  e.  ( C I B ) )
13 ismir.1 . . . 4  |-  ( ph  ->  C  e.  P )
141, 2, 3, 6, 9, 7mirreu3 23900 . . . 4  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
15 oveq2 6285 . . . . . . 7  |-  ( z  =  C  ->  ( A  .-  z )  =  ( A  .-  C
) )
1615eqeq1d 2443 . . . . . 6  |-  ( z  =  C  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  C )  =  ( A  .-  B ) ) )
17 oveq1 6284 . . . . . . 7  |-  ( z  =  C  ->  (
z I B )  =  ( C I B ) )
1817eleq2d 2511 . . . . . 6  |-  ( z  =  C  ->  ( A  e.  ( z
I B )  <->  A  e.  ( C I B ) ) )
1916, 18anbi12d 710 . . . . 5  |-  ( z  =  C  ->  (
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) )  <->  ( ( A 
.-  C )  =  ( A  .-  B
)  /\  A  e.  ( C I B ) ) ) )
2019riota2 6261 . . . 4  |-  ( ( C  e.  P  /\  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  ->  (
( ( A  .-  C )  =  ( A  .-  B )  /\  A  e.  ( C I B ) )  <->  ( iota_ z  e.  P  ( ( A 
.-  z )  =  ( A  .-  B
)  /\  A  e.  ( z I B ) ) )  =  C ) )
2113, 14, 20syl2anc 661 . . 3  |-  ( ph  ->  ( ( ( A 
.-  C )  =  ( A  .-  B
)  /\  A  e.  ( C I B ) )  <->  ( iota_ z  e.  P  ( ( A 
.-  z )  =  ( A  .-  B
)  /\  A  e.  ( z I B ) ) )  =  C ) )
2211, 12, 21mpbi2and 919 . 2  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  C )
2310, 22eqtr2d 2483 1  |-  ( ph  ->  C  =  ( M `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E!wreu 2793   ` cfv 5574   iota_crio 6237  (class class class)co 6277   Basecbs 14504   distcds 14578  TarskiGcstrkg 23690  Itvcitv 23697  LineGclng 23698  pInvGcmir 23898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-trkgc 23709  df-trkgb 23710  df-trkgcb 23711  df-trkg 23715  df-mir 23899
This theorem is referenced by:  mirmir  23908  mireq  23911  mirinv  23912  miriso  23915  mirmir2  23919  mirauto  23926  colmid  23930  krippenlem  23932  midexlem  23934  mideulem2  23973  opphllem  23974  midcom  24013
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