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Theorem tgbtwnconn1 24669
Description: Connectivity 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 774 . . . . . . . 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 465 . . . . . . 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 471 . . . . . 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 467 . . . . . . 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 6331 . . . . . 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 2543 . . . . 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 399 . . . 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 769 . . . . . . 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 471 . . . . . 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 467 . . . . . . 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 6331 . . . . . 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 2543 . . . . 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 398 . . . 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 2635 . . . . . 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 743 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 743 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 743 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 743 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 743 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 795 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 17 . . . . . . . . . . 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 17 . . . . . . . . . . 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 2462 . . . . . . . . . . 11  |-  ( dist `  G )  =  (
dist `  G )
40 simp-4r 782 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 767 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 786 . . . . . . . . . . 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 782 . . . . . . . . . . 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 749 . . . . . . . . . . 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 749 . . . . . . . . . . 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 784 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 774 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 469 . . . . . . . . . . . . 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 749 . . . . . . . . . . . 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 24573 . . . . . . . . . . 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 771 . . . . . . . . . . . 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 749 . . . . . . . . . . 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 469 . . . . . . . . . . 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 469 . . . . . . . . . . 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 767 . . . . . . . . . . 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 769 . . . . . . . . . . 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 771 . . . . . . . . . . 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 788 . . . . . . . . . . 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 24668 . . . . . . . . . 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 24581 . . . . . . . . . . . 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 24581 . . . . . . . . . . . 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 24564 . . . . . . . . . . 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 745 . . . . . . . . . 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 2944 . . . . . . . . 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 24561 . . . . . . . . . 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 741 . . . . . . . . 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 2944 . . . . . . . 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 24561 . . . . . . . . 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 471 . . . . . . . 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 2944 . . . . . . 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 440 . . . . . 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 226 . . . . 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 384 . . . 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 800 . . 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 24561 . . . 4  |-  ( ph  ->  E. f  e.  P  ( C  e.  ( A I f )  /\  ( C (
dist `  G )
f )  =  ( C ( dist `  G
) D ) ) )
8079ad2antrr 737 . . 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 2944 . 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 24561 . 2  |-  ( ph  ->  E. e  e.  P  ( D  e.  ( A I e )  /\  ( D (
dist `  G )
e )  =  ( D ( dist `  G
) C ) ) )
8381, 82r19.29a 2944 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 374    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750   ` cfv 5601  (class class class)co 6315   Basecbs 15170   distcds 15248  TarskiGcstrkg 24527  Itvcitv 24533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-hash 12548  df-word 12697  df-concat 12699  df-s1 12700  df-s2 12981  df-s3 12982  df-trkgc 24545  df-trkgb 24546  df-trkgcb 24547  df-trkg 24550  df-cgrg 24605
This theorem is referenced by:  tgbtwnconn2  24670  tgbtwnconnln1  24674  hltr  24704  hlbtwn  24705
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