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Theorem mireq 24710
Description: Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-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 )
mireq.c  |-  ( ph  ->  C  e.  P )
mireq.d  |-  ( ph  ->  ( M `  B
)  =  ( M `
 C ) )
Assertion
Ref Expression
mireq  |-  ( ph  ->  B  =  C )

Proof of Theorem mireq
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 mireq.c . . . 4  |-  ( ph  ->  C  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 24706 . . 3  |-  ( ph  ->  ( M `  C
)  e.  P )
11 mirmir.b . . 3  |-  ( ph  ->  B  e.  P )
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 24701 . . . . . . 7  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
13 mireq.d . . . . . . 7  |-  ( ph  ->  ( M `  B
)  =  ( M `
 C ) )
1412, 13eqtr3d 2487 . . . . . 6  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  ( M `  C ) )
151, 2, 3, 6, 11, 7mirreu3 24699 . . . . . . 7  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
16 oveq2 6298 . . . . . . . . . 10  |-  ( z  =  ( M `  C )  ->  ( A  .-  z )  =  ( A  .-  ( M `  C )
) )
1716eqeq1d 2453 . . . . . . . . 9  |-  ( z  =  ( M `  C )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  C
) )  =  ( A  .-  B ) ) )
18 oveq1 6297 . . . . . . . . . 10  |-  ( z  =  ( M `  C )  ->  (
z I B )  =  ( ( M `
 C ) I B ) )
1918eleq2d 2514 . . . . . . . . 9  |-  ( z  =  ( M `  C )  ->  ( A  e.  ( z
I B )  <->  A  e.  ( ( M `  C ) I B ) ) )
2017, 19anbi12d 717 . . . . . . . 8  |-  ( z  =  ( M `  C )  ->  (
( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) )  <->  ( ( A 
.-  ( M `  C ) )  =  ( A  .-  B
)  /\  A  e.  ( ( M `  C ) I B ) ) ) )
2120riota2 6274 . . . . . . 7  |-  ( ( ( M `  C
)  e.  P  /\  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  ->  (
( ( A  .-  ( M `  C ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  C )
I B ) )  <-> 
( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  ( M `  C ) ) )
2210, 15, 21syl2anc 667 . . . . . 6  |-  ( ph  ->  ( ( ( A 
.-  ( M `  C ) )  =  ( A  .-  B
)  /\  A  e.  ( ( M `  C ) I B ) )  <->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  ( M `  C
) ) )
2314, 22mpbird 236 . . . . 5  |-  ( ph  ->  ( ( A  .-  ( M `  C ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  C )
I B ) ) )
2423simpld 461 . . . 4  |-  ( ph  ->  ( A  .-  ( M `  C )
)  =  ( A 
.-  B ) )
2524eqcomd 2457 . . 3  |-  ( ph  ->  ( A  .-  B
)  =  ( A 
.-  ( M `  C ) ) )
2623simprd 465 . . . 4  |-  ( ph  ->  A  e.  ( ( M `  C ) I B ) )
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 24532 . . 3  |-  ( ph  ->  A  e.  ( B I ( M `  C ) ) )
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 24704 . 2  |-  ( ph  ->  B  =  ( M `
 ( M `  C ) ) )
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 24707 . 2  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
3028, 29eqtrd 2485 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E!wreu 2739   ` cfv 5582   iota_crio 6251  (class class class)co 6290   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485  pInvGcmir 24697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501  df-mir 24698
This theorem is referenced by:  mirhl  24724  mirbtwnhl  24725  mirhl2  24726  colperpexlem3  24774
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