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Theorem tgbtwnexch3 23075
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 23069 . 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 23069 . 2  |-  ( ph  ->  C  e.  ( D I A ) )
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 23074 1  |-  ( ph  ->  C  e.  ( B I D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   distcds 14358  TarskiGcstrkg 23015  Itvcitv 23022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-trkgc 23034  df-trkgb 23035  df-trkgcb 23036  df-trkg 23040
This theorem is referenced by:  tgbtwnouttr2  23076  tgbtwnexch2  23077  tgifscgr  23090  tgcgrxfr  23099  tgbtwnconn1lem1  23134  tgbtwnconn1lem2  23135  tgbtwnconn1lem3  23136  tgbtwnconn2  23138  tgbtwnconn3  23139  tglineeltr  23169  miriso  23209  krippenlem  23220
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