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Theorem ismir 23214
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 23211 . 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 23209 . . . 4  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
15 oveq2 6211 . . . . . . 7  |-  ( z  =  C  ->  ( A  .-  z )  =  ( A  .-  C
) )
1615eqeq1d 2456 . . . . . 6  |-  ( z  =  C  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  C )  =  ( A  .-  B ) ) )
17 oveq1 6210 . . . . . . 7  |-  ( z  =  C  ->  (
z I B )  =  ( C I B ) )
1817eleq2d 2524 . . . . . 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 6187 . . . 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 912 . 2  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  C )
2310, 22eqtr2d 2496 1  |-  ( ph  ->  C  =  ( M `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E!wreu 2801   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   distcds 14370  TarskiGcstrkg 23032  Itvcitv 23039  LineGclng 23040  pInvGcmir 23207
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-trkgc 23051  df-trkgb 23052  df-trkgcb 23053  df-trkg 23057  df-mir 23208
This theorem is referenced by:  mirmir  23217  mireq  23220  mirinv  23221  miriso  23224  mirmir2  23228  mirauto  23229  colmid  23233  krippenlem  23235  midexlem  23237  mideulem  23270  midcom  23282
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