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Theorem mirreu 24430
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 24427 . 2  |-  ( ph  ->  ( M `  B
)  e.  P )
111, 2, 3, 4, 5, 6, 7, 8, 9mirmir 24428 . 2  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
126ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  G  e. TarskiG )
137ad2antrr 724 . . . . . 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 24428 . . . . 5  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  ( M `  a ) )  =  a )
16 simpr 459 . . . . . 6  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  a )  =  B )
1716fveq2d 5853 . . . . 5  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  ( M `  ( M `  a ) )  =  ( M `  B
) )
1815, 17eqtr3d 2445 . . . 4  |-  ( ( ( ph  /\  a  e.  P )  /\  ( M `  a )  =  B )  ->  a  =  ( M `  B ) )
1918ex 432 . . 3  |-  ( (
ph  /\  a  e.  P )  ->  (
( M `  a
)  =  B  -> 
a  =  ( M `
 B ) ) )
2019ralrimiva 2818 . 2  |-  ( ph  ->  A. a  e.  P  ( ( M `  a )  =  B  ->  a  =  ( M `  B ) ) )
21 fveq2 5849 . . . 4  |-  ( a  =  ( M `  B )  ->  ( M `  a )  =  ( M `  ( M `  B ) ) )
2221eqeq1d 2404 . . 3  |-  ( a  =  ( M `  B )  ->  (
( M `  a
)  =  B  <->  ( M `  ( M `  B
) )  =  B ) )
2322eqreu 3241 . 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 1230 1  |-  ( ph  ->  E! a  e.  P  ( M `  a )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E!wreu 2756   ` cfv 5569   Basecbs 14841   distcds 14918  TarskiGcstrkg 24206  Itvcitv 24212  LineGclng 24213  pInvGcmir 24418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-trkgc 24224  df-trkgb 24225  df-trkgcb 24226  df-trkg 24229  df-mir 24419
This theorem is referenced by: (None)
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