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Theorem tgbtwnconn1 23827
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 758 . . . . . . . 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 6311 . . . . . 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 2558 . . . . 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 755 . . . . . . 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 6311 . . . . . 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 2558 . . . . 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 2664 . . . . . 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 779 . . . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( dist `  G )  =  (
dist `  G )
40 simp-4r 766 . . . . . . . . . . . 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 754 . . . . . . . . . . . 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 770 . . . . . . . . . . 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 766 . . . . . . . . . . 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 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 ) ) )  /\  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 758 . . . . . . . . . . . 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 23737 . . . . . . . . . . 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 756 . . . . . . . . . . . 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 754 . . . . . . . . . . 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 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.  ( C I e ) )
61 simprr 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.  ( D I f ) )
62 simp-7r 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 ) ) )  ->  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 23826 . . . . . . . . . 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 23745 . . . . . . . . . . . 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 23745 . . . . . . . . . . . 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 23730 . . . . . . . . . . 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 3008 . . . . . . . . 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 23727 . . . . . . . . . 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 3008 . . . . . . . 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 23727 . . . . . . . . 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 3008 . . . . . . 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 784 . . 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 23727 . . . 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 3008 . 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 23727 . 2  |-  ( ph  ->  E. e  e.  P  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )
8381, 82r19.29a 3008 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   ` cfv 5594  (class class class)co 6295   Basecbs 14507   distcds 14581  TarskiGcstrkg 23691  Itvcitv 23698
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-s2 12793  df-s3 12794  df-trkgc 23710  df-trkgb 23711  df-trkgcb 23712  df-trkg 23716  df-cgrg 23769
This theorem is referenced by:  tgbtwnconn2  23828  tgbtwnconnln1  23832
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