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Theorem ismir 24697
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 24694 . 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 24692 . . . 4  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
15 oveq2 6296 . . . . . . 7  |-  ( z  =  C  ->  ( A  .-  z )  =  ( A  .-  C
) )
1615eqeq1d 2452 . . . . . 6  |-  ( z  =  C  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  C )  =  ( A  .-  B ) ) )
17 oveq1 6295 . . . . . . 7  |-  ( z  =  C  ->  (
z I B )  =  ( C I B ) )
1817eleq2d 2513 . . . . . 6  |-  ( z  =  C  ->  ( A  e.  ( z
I B )  <->  A  e.  ( C I B ) ) )
1916, 18anbi12d 716 . . . . 5  |-  ( z  =  C  ->  (
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) )  <->  ( ( A 
.-  C )  =  ( A  .-  B
)  /\  A  e.  ( C I B ) ) ) )
2019riota2 6272 . . . 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 666 . . 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 931 . 2  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  C )
2310, 22eqtr2d 2485 1  |-  ( ph  ->  C  =  ( M `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   E!wreu 2738   ` cfv 5581   iota_crio 6249  (class class class)co 6288   Basecbs 15114   distcds 15192  TarskiGcstrkg 24471  Itvcitv 24477  LineGclng 24478  pInvGcmir 24690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-trkgc 24489  df-trkgb 24490  df-trkgcb 24491  df-trkg 24494  df-mir 24691
This theorem is referenced by:  mirmir  24700  mireq  24703  mirinv  24704  miriso  24708  mirmir2  24712  mirauto  24722  colmid  24726  krippenlem  24728  midexlem  24730  mideulem2  24769  opphllem  24770  midcom  24817  trgcopyeulem  24840
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