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Theorem mircom 23785
Description: Variation on mirmir 23784 (Contributed by Thierry Arnoux, 10-Nov-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 )
mircom.1  |-  ( ph  ->  ( M `  B
)  =  C )
Assertion
Ref Expression
mircom  |-  ( ph  ->  ( M `  C
)  =  B )

Proof of Theorem mircom
StepHypRef Expression
1 mircom.1 . . 3  |-  ( ph  ->  ( M `  B
)  =  C )
21fveq2d 5870 . 2  |-  ( ph  ->  ( M `  ( M `  B )
)  =  ( M `
 C ) )
3 mirval.p . . 3  |-  P  =  ( Base `  G
)
4 mirval.d . . 3  |-  .-  =  ( dist `  G )
5 mirval.i . . 3  |-  I  =  (Itv `  G )
6 mirval.l . . 3  |-  L  =  (LineG `  G )
7 mirval.s . . 3  |-  S  =  (pInvG `  G )
8 mirval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
9 mirval.a . . 3  |-  ( ph  ->  A  e.  P )
10 mirfv.m . . 3  |-  M  =  ( S `  A
)
11 mirmir.b . . 3  |-  ( ph  ->  B  e.  P )
123, 4, 5, 6, 7, 8, 9, 10, 11mirmir 23784 . 2  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
132, 12eqtr3d 2510 1  |-  ( ph  ->  ( M `  C
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5588   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  pInvGcmir 23774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-mir 23775
This theorem is referenced by:  colperpexlem3  23839  mideulem  23841  mideu  23842
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