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Theorem ismir 23868
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 23865 . 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 23863 . . . 4  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
15 oveq2 6302 . . . . . . 7  |-  ( z  =  C  ->  ( A  .-  z )  =  ( A  .-  C
) )
1615eqeq1d 2469 . . . . . 6  |-  ( z  =  C  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  C )  =  ( A  .-  B ) ) )
17 oveq1 6301 . . . . . . 7  |-  ( z  =  C  ->  (
z I B )  =  ( C I B ) )
1817eleq2d 2537 . . . . . 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 6278 . . . 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 2509 1  |-  ( ph  ->  C  =  ( M `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2819   ` cfv 5593   iota_crio 6254  (class class class)co 6294   Basecbs 14502   distcds 14576  TarskiGcstrkg 23668  Itvcitv 23675  LineGclng 23676  pInvGcmir 23861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-trkgc 23687  df-trkgb 23688  df-trkgcb 23689  df-trkg 23693  df-mir 23862
This theorem is referenced by:  mirmir  23871  mireq  23874  mirinv  23875  miriso  23878  mirmir2  23882  mirauto  23884  colmid  23888  krippenlem  23890  midexlem  23892  mideulem  23928  midcom  23940
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