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Theorem mirreu 23753
Description: Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (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
)
mirmir.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
mirreu  |-  ( ph  ->  E! a  e.  P  ( M `  a )  =  B )
Distinct variable groups:    B, a    M, a    P, a    ph, a
Allowed substitution hints:    A( a)    S( a)    G( a)    I( a)    L( a)    .- ( a)

Proof of Theorem mirreu
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 mirmir.b . . 3  |-  ( ph  ->  B  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 23750 . 2  |-  ( ph  ->  ( M `  B
)  e.  P )
111, 2, 3, 4, 5, 6, 7, 8, 9mirmir 23751 . 2  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
126ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  G  e. TarskiG )
137ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  A  e.  P )
14 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  a  e.  P )
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 23751 . . . . 5  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  ( M `  a ) )  =  a )
16 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  a )  =  B )
1716fveq2d 5863 . . . . 5  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  ( M `  a ) )  =  ( M `  B
) )
1815, 17eqtr3d 2505 . . . 4  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  a  =  ( M `  B ) )
1918ex 434 . . 3  |-  ( (
ph  /\  a  e.  P )  ->  (
( M `  a
)  =  B  -> 
a  =  ( M `
 B ) ) )
2019ralrimiva 2873 . 2  |-  ( ph  ->  A. a  e.  P  ( ( M `  a )  =  B  ->  a  =  ( M `  B ) ) )
21 fveq2 5859 . . . 4  |-  ( a  =  ( M `  B )  ->  ( M `  a )  =  ( M `  ( M `  B ) ) )
2221eqeq1d 2464 . . 3  |-  ( a  =  ( M `  B )  ->  (
( M `  a
)  =  B  <->  ( M `  ( M `  B
) )  =  B ) )
2322eqreu 3290 . 2  |-  ( ( ( M `  B
)  e.  P  /\  ( M `  ( M `
 B ) )  =  B  /\  A. a  e.  P  (
( M `  a
)  =  B  -> 
a  =  ( M `
 B ) ) )  ->  E! a  e.  P  ( M `  a )  =  B )
2410, 11, 20, 23syl3anc 1223 1  |-  ( ph  ->  E! a  e.  P  ( M `  a )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E!wreu 2811   ` cfv 5581   Basecbs 14481   distcds 14555  TarskiGcstrkg 23548  Itvcitv 23555  LineGclng 23556  pInvGcmir 23741
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-trkgc 23567  df-trkgb 23568  df-trkgcb 23569  df-trkg 23573  df-mir 23742
This theorem is referenced by: (None)
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