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Theorem mircom 24565
Description: Variation on mirmir 24564 (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 5876 . 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 24564 . 2  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
132, 12eqtr3d 2463 1  |-  ( ph  ->  ( M `  C
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   ` cfv 5592   Basecbs 15073   distcds 15151  TarskiGcstrkg 24338  Itvcitv 24344  LineGclng 24345  pInvGcmir 24554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-trkgc 24356  df-trkgb 24357  df-trkgcb 24358  df-trkg 24361  df-mir 24555
This theorem is referenced by:  miduniq  24584  colperpexlem3  24628  mideulem2  24630  midex  24633  opphllem1  24643  opphllem2  24644  opphllem3  24645  opphllem5  24647  opphllem6  24648  trgcopyeulem  24700
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