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Theorem tgbtwncom 22942
Description: Betweeness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-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 )
tgbtwntriv2.1  |-  ( ph  ->  A  e.  P )
tgbtwntriv2.2  |-  ( ph  ->  B  e.  P )
tgbtwncom.3  |-  ( ph  ->  C  e.  P )
tgbtwncom.4  |-  ( ph  ->  B  e.  ( A I C ) )
Assertion
Ref Expression
tgbtwncom  |-  ( ph  ->  B  e.  ( C I A ) )

Proof of Theorem tgbtwncom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4  |-  P  =  ( Base `  G
)
2 tkgeom.d . . . 4  |-  .-  =  ( dist `  G )
3 tkgeom.i . . . 4  |-  I  =  (Itv `  G )
4 tkgeom.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  G  e. TarskiG )
6 tgbtwntriv2.2 . . . . 5  |-  ( ph  ->  B  e.  P )
76ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  B  e.  P
)
8 simplr 754 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  x  e.  P
)
9 simprl 755 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  x  e.  ( B I B ) )
101, 2, 3, 5, 7, 8, 9axtgbtwnid 22927 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  B  =  x )
11 simprr 756 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  x  e.  ( C I A ) )
1210, 11eqeltrd 2517 . 2  |-  ( ( ( ph  /\  x  e.  P )  /\  (
x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )  ->  B  e.  ( C I A ) )
13 tgbtwntriv2.1 . . 3  |-  ( ph  ->  A  e.  P )
14 tgbtwncom.3 . . 3  |-  ( ph  ->  C  e.  P )
15 tgbtwncom.4 . . 3  |-  ( ph  ->  B  e.  ( A I C ) )
161, 2, 3, 4, 6, 14tgbtwntriv2 22941 . . 3  |-  ( ph  ->  C  e.  ( B I C ) )
171, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16axtgpasch 22928 . 2  |-  ( ph  ->  E. x  e.  P  ( x  e.  ( B I B )  /\  x  e.  ( C I A ) ) )
1812, 17r19.29a 2862 1  |-  ( ph  ->  B  e.  ( C I A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-trkgc 22909  df-trkgb 22910  df-trkgcb 22911  df-trkg 22916
This theorem is referenced by:  tgbtwncomb  22943  tgbtwntriv1  22944  tgbtwnexch3  22947  tgbtwnexch2  22949  tgbtwnouttr  22950  tgbtwnexch  22951  tgtrisegint  22952  tgifscgr  22961  tgcgrxfr  22970  tgbtwnconn1lem1  23004  tgbtwnconn1lem2  23005  tgbtwnconn1lem3  23006  tgbtwnconn1  23007  tgbtwnconn3  23009  tgbtwnconnln1  23011  tgbtwnconnln2  23012  legtri3  23021  legtrid  23022  legbtwn  23025  tglineeltr  23037  mirreu3  23058  mirmir  23066  mireq  23069  miriso  23073  miduniq  23079  colmid  23082  krippenlem  23084  krippen  23085  midexlem  23086  ragflat  23098  ragcgr  23101  footex  23109  colperplem1  23112
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