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Theorem tgcgrcomlr 23599
Description: Congruence commutes on both sides. Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgcgrcomlr.a  |-  ( ph  ->  A  e.  P )
tgcgrcomlr.b  |-  ( ph  ->  B  e.  P )
tgcgrcomlr.c  |-  ( ph  ->  C  e.  P )
tgcgrcomlr.d  |-  ( ph  ->  D  e.  P )
tgcgrcomlr.6  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
Assertion
Ref Expression
tgcgrcomlr  |-  ( ph  ->  ( B  .-  A
)  =  ( D 
.-  C ) )

Proof of Theorem tgcgrcomlr
StepHypRef Expression
1 tgcgrcomlr.6 . 2  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
2 tkgeom.p . . 3  |-  P  =  ( Base `  G
)
3 tkgeom.d . . 3  |-  .-  =  ( dist `  G )
4 tkgeom.i . . 3  |-  I  =  (Itv `  G )
5 tkgeom.g . . 3  |-  ( ph  ->  G  e. TarskiG )
6 tgcgrcomlr.a . . 3  |-  ( ph  ->  A  e.  P )
7 tgcgrcomlr.b . . 3  |-  ( ph  ->  B  e.  P )
82, 3, 4, 5, 6, 7axtgcgrrflx 23587 . 2  |-  ( ph  ->  ( A  .-  B
)  =  ( B 
.-  A ) )
9 tgcgrcomlr.c . . 3  |-  ( ph  ->  C  e.  P )
10 tgcgrcomlr.d . . 3  |-  ( ph  ->  D  e.  P )
112, 3, 4, 5, 9, 10axtgcgrrflx 23587 . 2  |-  ( ph  ->  ( C  .-  D
)  =  ( D 
.-  C ) )
121, 8, 113eqtr3d 2516 1  |-  ( ph  ->  ( B  .-  A
)  =  ( D 
.-  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   distcds 14560  TarskiGcstrkg 23553  Itvcitv 23560
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-trkgc 23572  df-trkg 23578
This theorem is referenced by:  tgcgrextend  23604  tgifscgr  23628  tgcgrsub  23629  trgcgrg  23634  tgcgrxfr  23637  cgr3swap12  23642  cgr3swap23  23643  tgbtwnxfr  23646  lnext  23681  tgbtwnconn1lem1  23686  tgbtwnconn1lem2  23687  tgbtwnconn1lem3  23688  tgbtwnconn1  23689  legov2  23700  legtri3  23704  legbtwn  23708  miriso  23763  miduniq  23770  colmid  23773  symquadlem  23774  krippenlem  23775  midexlem  23777  ragcom  23783  ragflat  23789  ragcgr  23792  footex  23803  colperpexlem1  23809  mideulem  23813  lmiisolem  23838  hypcgrlem1  23841
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