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Theorem mireq 24247
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 24243 . . 3  |-  ( ph  ->  ( M `  C
)  e.  P )
11 mirmir.b . . 3  |-  ( ph  ->  B  e.  P )
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 24238 . . . . . . 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 2497 . . . . . 6  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  =  ( M `  C ) )
151, 2, 3, 6, 11, 7mirreu3 24236 . . . . . . 7  |-  ( ph  ->  E! z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )
16 oveq2 6278 . . . . . . . . . 10  |-  ( z  =  ( M `  C )  ->  ( A  .-  z )  =  ( A  .-  ( M `  C )
) )
1716eqeq1d 2456 . . . . . . . . 9  |-  ( z  =  ( M `  C )  ->  (
( A  .-  z
)  =  ( A 
.-  B )  <->  ( A  .-  ( M `  C
) )  =  ( A  .-  B ) ) )
18 oveq1 6277 . . . . . . . . . 10  |-  ( z  =  ( M `  C )  ->  (
z I B )  =  ( ( M `
 C ) I B ) )
1918eleq2d 2524 . . . . . . . . 9  |-  ( z  =  ( M `  C )  ->  ( A  e.  ( z
I B )  <->  A  e.  ( ( M `  C ) I B ) ) )
2017, 19anbi12d 708 . . . . . . . 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 6254 . . . . . . 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 659 . . . . . 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 232 . . . . 5  |-  ( ph  ->  ( ( A  .-  ( M `  C ) )  =  ( A 
.-  B )  /\  A  e.  ( ( M `  C )
I B ) ) )
2423simpld 457 . . . 4  |-  ( ph  ->  ( A  .-  ( M `  C )
)  =  ( A 
.-  B ) )
2524eqcomd 2462 . . 3  |-  ( ph  ->  ( A  .-  B
)  =  ( A 
.-  ( M `  C ) ) )
2623simprd 461 . . . 4  |-  ( ph  ->  A  e.  ( ( M `  C ) I B ) )
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 24080 . . 3  |-  ( ph  ->  A  e.  ( B I ( M `  C ) ) )
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 24241 . 2  |-  ( ph  ->  B  =  ( M `
 ( M `  C ) ) )
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 24244 . 2  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
3028, 29eqtrd 2495 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-mir 24235
This theorem is referenced by:  mirhl  24260  mirbtwnhl  24261  colperpexlem3  24307
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