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Theorem tgbtwnconn1 24088
Description: Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Hypotheses
Ref Expression
tgbtwnconn1.p  |-  P  =  ( Base `  G
)
tgbtwnconn1.i  |-  I  =  (Itv `  G )
tgbtwnconn1.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnconn1.a  |-  ( ph  ->  A  e.  P )
tgbtwnconn1.b  |-  ( ph  ->  B  e.  P )
tgbtwnconn1.c  |-  ( ph  ->  C  e.  P )
tgbtwnconn1.d  |-  ( ph  ->  D  e.  P )
tgbtwnconn1.1  |-  ( ph  ->  A  =/=  B )
tgbtwnconn1.2  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnconn1.3  |-  ( ph  ->  B  e.  ( A I D ) )
Assertion
Ref Expression
tgbtwnconn1  |-  ( ph  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )

Proof of Theorem tgbtwnconn1
Dummy variables  e 
f  h  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 760 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )
21simpld 459 . . . . . . 7  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  D  e.  ( A I e ) )
32adantr 465 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =  e )  ->  D  e.  ( A I e ) )
4 simpr 461 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =  e )  ->  C  =  e )
54oveq2d 6312 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =  e )  -> 
( A I C )  =  ( A I e ) )
63, 5eleqtrrd 2548 . . . . 5  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =  e )  ->  D  e.  ( A I C ) )
76olcd 393 . . . 4  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =  e )  -> 
( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
8 simprl 756 . . . . . . 7  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  C  e.  ( A I f ) )
98adantr 465 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  D  =  f )  ->  C  e.  ( A I f ) )
10 simpr 461 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  D  =  f )  ->  D  =  f )
1110oveq2d 6312 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  D  =  f )  -> 
( A I D )  =  ( A I f ) )
129, 11eleqtrrd 2548 . . . . 5  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  D  =  f )  ->  C  e.  ( A I D ) )
1312orcd 392 . . . 4  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  D  =  f )  -> 
( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
14 df-ne 2654 . . . . . 6  |-  ( C  =/=  e  <->  -.  C  =  e )
15 tgbtwnconn1.p . . . . . . . . . . 11  |-  P  =  ( Base `  G
)
16 tgbtwnconn1.i . . . . . . . . . . 11  |-  I  =  (Itv `  G )
17 tgbtwnconn1.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e. TarskiG )
1817ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  G  e. TarskiG )
1918ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  G  e. TarskiG )
20 tgbtwnconn1.a . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  P )
2120ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  A  e.  P )
2221ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  A  e.  P )
23 tgbtwnconn1.b . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  P )
2423ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  B  e.  P )
2524ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  B  e.  P )
26 tgbtwnconn1.c . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  P )
2726ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  C  e.  P )
2827ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  C  e.  P )
29 tgbtwnconn1.d . . . . . . . . . . . . 13  |-  ( ph  ->  D  e.  P )
3029ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  D  e.  P )
3130ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  D  e.  P )
32 simp-11l 781 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ph )
33 tgbtwnconn1.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  =/=  B )
3432, 33syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  A  =/=  B )
35 tgbtwnconn1.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ( A I C ) )
3632, 35syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  B  e.  ( A I C ) )
37 tgbtwnconn1.3 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ( A I D ) )
3832, 37syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  B  e.  ( A I D ) )
39 eqid 2457 . . . . . . . . . . 11  |-  ( dist `  G )  =  (
dist `  G )
40 simp-4r 768 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  e  e.  P )
4140ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  e  e.  P )
42 simplr 755 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  f  e.  P )
4342ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  f  e.  P )
44 simp-6r 772 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  h  e.  P )
45 simp-4r 768 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  j  e.  P )
462ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  D  e.  ( A I e ) )
478ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  C  e.  ( A I f ) )
48 simp-5r 770 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( e  e.  ( A I h )  /\  ( e ( dist `  G
) h )  =  ( B ( dist `  G ) C ) ) )
4948simpld 459 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  e  e.  ( A I h ) )
50 simpllr 760 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )
5150simpld 459 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  f  e.  ( A I j ) )
521simprd 463 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) )
5352ad7antr 737 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( D
( dist `  G )
e )  =  ( D ( dist `  G
) C ) )
5415, 39, 16, 19, 31, 41, 31, 28, 53tgcgrcomlr 23997 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( e
( dist `  G ) D )  =  ( C ( dist `  G
) D ) )
55 simprr 757 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) )
5655ad7antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( C
( dist `  G )
f )  =  ( C ( dist `  G
) D ) )
5748simprd 463 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( e
( dist `  G )
h )  =  ( B ( dist `  G
) C ) )
5850simprd 463 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  ( f
( dist `  G )
j )  =  ( B ( dist `  G
) D ) )
59 simplr 755 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  x  e.  P )
60 simprl 756 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  x  e.  ( C I e ) )
61 simprr 757 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  x  e.  ( D I f ) )
62 simp-7r 774 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  C  =/=  e )
6315, 16, 19, 22, 25, 28, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 54, 56, 57, 58, 59, 60, 61, 62tgbtwnconn1lem3 24087 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  /\  x  e.  P )  /\  (
x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )  ->  D  =  f )
6415, 39, 16, 18, 21, 27, 42, 8tgbtwncom 24005 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  C  e.  ( f I A ) )
6515, 39, 16, 18, 21, 30, 40, 2tgbtwncom 24005 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  D  e.  ( e I A ) )
6615, 39, 16, 18, 42, 40, 21, 27, 30, 64, 65axtgpasch 23990 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  E. x  e.  P  ( x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )
6766ad5antr 733 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  ->  E. x  e.  P  ( x  e.  ( C I e )  /\  x  e.  ( D I f ) ) )
6863, 67r19.29a 2999 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  /\  j  e.  P
)  /\  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )  ->  D  =  f )
6915, 39, 16, 18, 21, 42, 24, 30axtgsegcon 23987 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  E. j  e.  P  ( f  e.  ( A I j )  /\  ( f ( dist `  G
) j )  =  ( B ( dist `  G ) D ) ) )
7069ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  ->  E. j  e.  P  ( f  e.  ( A I j )  /\  ( f (
dist `  G )
j )  =  ( B ( dist `  G
) D ) ) )
7168, 70r19.29a 2999 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  /\  h  e.  P )  /\  (
e  e.  ( A I h )  /\  ( e ( dist `  G ) h )  =  ( B (
dist `  G ) C ) ) )  ->  D  =  f )
7215, 39, 16, 18, 21, 40, 24, 27axtgsegcon 23987 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  E. h  e.  P  ( e  e.  ( A I h )  /\  ( e ( dist `  G
) h )  =  ( B ( dist `  G ) C ) ) )
7372adantr 465 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  ->  E. h  e.  P  ( e  e.  ( A I h )  /\  ( e ( dist `  G
) h )  =  ( B ( dist `  G ) C ) ) )
7471, 73r19.29a 2999 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  /\  C  =/=  e )  ->  D  =  f )
7574ex 434 . . . . . 6  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( C  =/=  e  ->  D  =  f ) )
7614, 75syl5bir 218 . . . . 5  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( -.  C  =  e  ->  D  =  f ) )
7776orrd 378 . . . 4  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( C  =  e  \/  D  =  f )
)
787, 13, 77mpjaodan 786 . . 3  |-  ( ( ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  /\  f  e.  P )  /\  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
7915, 39, 16, 17, 20, 26, 26, 29axtgsegcon 23987 . . . 4  |-  ( ph  ->  E. f  e.  P  ( C  e.  ( A I f )  /\  ( C (
dist `  G )
f )  =  ( C ( dist `  G
) D ) ) )
8079ad2antrr 725 . . 3  |-  ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  ->  E. f  e.  P  ( C  e.  ( A I f )  /\  ( C ( dist `  G
) f )  =  ( C ( dist `  G ) D ) ) )
8178, 80r19.29a 2999 . 2  |-  ( ( ( ph  /\  e  e.  P )  /\  ( D  e.  ( A I e )  /\  ( D ( dist `  G
) e )  =  ( D ( dist `  G ) C ) ) )  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
8215, 39, 16, 17, 20, 29, 29, 26axtgsegcon 23987 . 2  |-  ( ph  ->  E. e  e.  P  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )
8381, 82r19.29a 2999 1  |-  ( ph  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   ` cfv 5594  (class class class)co 6296   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029
This theorem is referenced by:  tgbtwnconn2  24089  tgbtwnconnln1  24093  hltr  24120  hlbtwn  24121
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