MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnexch3 Structured version   Unicode version

Theorem tgbtwnexch3 23613
Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-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 )
tgbtwnintr.1  |-  ( ph  ->  A  e.  P )
tgbtwnintr.2  |-  ( ph  ->  B  e.  P )
tgbtwnintr.3  |-  ( ph  ->  C  e.  P )
tgbtwnintr.4  |-  ( ph  ->  D  e.  P )
tgbtwnexch3.5  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnexch3.6  |-  ( ph  ->  C  e.  ( A I D ) )
Assertion
Ref Expression
tgbtwnexch3  |-  ( ph  ->  C  e.  ( B I D ) )

Proof of Theorem tgbtwnexch3
StepHypRef Expression
1 tkgeom.p . 2  |-  P  =  ( Base `  G
)
2 tkgeom.d . 2  |-  .-  =  ( dist `  G )
3 tkgeom.i . 2  |-  I  =  (Itv `  G )
4 tkgeom.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwnintr.2 . 2  |-  ( ph  ->  B  e.  P )
6 tgbtwnintr.3 . 2  |-  ( ph  ->  C  e.  P )
7 tgbtwnintr.4 . 2  |-  ( ph  ->  D  e.  P )
8 tgbtwnintr.1 . 2  |-  ( ph  ->  A  e.  P )
9 tgbtwnexch3.5 . . 3  |-  ( ph  ->  B  e.  ( A I C ) )
101, 2, 3, 4, 8, 5, 6, 9tgbtwncom 23607 . 2  |-  ( ph  ->  B  e.  ( C I A ) )
11 tgbtwnexch3.6 . . 3  |-  ( ph  ->  C  e.  ( A I D ) )
121, 2, 3, 4, 8, 6, 7, 11tgbtwncom 23607 . 2  |-  ( ph  ->  C  e.  ( D I A ) )
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 23612 1  |-  ( ph  ->  C  e.  ( B I D ) )
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-pw 4012  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-trkgb 23573  df-trkgcb 23574  df-trkg 23578
This theorem is referenced by:  tgbtwnouttr2  23614  tgbtwnexch2  23615  tgifscgr  23628  tgcgrxfr  23637  tgbtwnconn1lem1  23686  tgbtwnconn1lem2  23687  tgbtwnconn1lem3  23688  tgbtwnconn2  23690  tgbtwnconn3  23691  tglineeltr  23725  miriso  23763  krippenlem  23775
  Copyright terms: Public domain W3C validator