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Theorem mirinv 23070
Description: The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49. , Theorem 7.10 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-Jul-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
)
mirinv.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
mirinv  |-  ( ph  ->  ( ( M `  B )  =  B  <-> 
A  =  B ) )

Proof of Theorem mirinv
StepHypRef Expression
1 mirval.p . . . 4  |-  P  =  ( Base `  G
)
2 mirval.d . . . 4  |-  .-  =  ( dist `  G )
3 mirval.i . . . 4  |-  I  =  (Itv `  G )
4 mirval.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  G  e. TarskiG )
6 mirinv.b . . . . 5  |-  ( ph  ->  B  e.  P )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  B  e.  P )
8 mirval.a . . . . 5  |-  ( ph  ->  A  e.  P )
98adantr 465 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  e.  P )
10 mirval.l . . . . . 6  |-  L  =  (LineG `  G )
11 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
12 mirfv.m . . . . . 6  |-  M  =  ( S `  A
)
131, 2, 3, 10, 11, 5, 9, 12, 7mirbtwn 23062 . . . . 5  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  e.  ( ( M `  B ) I B ) )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  ( M `  B )  =  B )
1514oveq1d 6106 . . . . 5  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  ( ( M `  B )
I B )  =  ( B I B ) )
1613, 15eleqtrd 2519 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  e.  ( B I B ) )
171, 2, 3, 5, 7, 9, 16axtgbtwnid 22927 . . 3  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  B  =  A )
1817eqcomd 2448 . 2  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  =  B )
194adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
208adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
216adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  B  e.  P )
22 eqidd 2444 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( A  .-  B
) )
23 simpr 461 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
241, 2, 3, 19, 21, 21tgbtwntriv1 22944 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  B  e.  ( B I B ) )
2523, 24eqeltrd 2517 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  e.  ( B I B ) )
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 23063 . . 3  |-  ( (
ph  /\  A  =  B )  ->  B  =  ( M `  B ) )
2726eqcomd 2448 . 2  |-  ( (
ph  /\  A  =  B )  ->  ( M `  B )  =  B )
2818, 27impbida 828 1  |-  ( ph  ->  ( ( M `  B )  =  B  <-> 
A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897  LineGclng 22898  pInvGcmir 23055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-trkgc 22909  df-trkgb 22910  df-trkgcb 22911  df-trkg 22916  df-mir 23056
This theorem is referenced by:  mircinv  23071  miduniq  23079  miduniq2  23081  krippenlem  23084  ragflat2  23097  footex  23109  colperplem2  23113
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