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Theorem segconeq 28053
Description: Two points that satsify the conclusion of axsegcon 23185 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
segconeq |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
Proof of Theorem segconeq
StepHypRef Expression
1 simpr2l 1047 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> A Btwn <. Q , X >. )
21, 1jca 532 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( A Btwn <. Q , X >. /\ A Btwn <. Q , X >. ) )
3 simpl1 991 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> N e. NN )
4 simpl31 1069 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> Q e. ( EE ` N ) )
5 simpl21 1066 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> A e. ( EE ` N ) )
63, 4, 5cgrrflxd 28031 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. Q , A >.Cgr <. Q , A >. )
7 simpl32 1070 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> X e. ( EE ` N ) )
83, 5, 7cgrrflxd 28031 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. A , X >. )
96, 8jca 532 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( <. Q , A >.Cgr <. Q , A >. /\ <. A , X >.Cgr <. A , X >. ) )
10 simpl33 1071 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> Y e. ( EE ` N ) )
114, 5, 103jca 1168 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ Y e. ( EE ` N ) ) )
124, 5, 73jca 1168 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ X e. ( EE ` N ) ) )
133, 11, 123jca 1168 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ Y e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) )
14 simpr3l 1049 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> A Btwn <. Q , Y >. )
1514, 1jca 532 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( A Btwn <. Q , Y >. /\ A Btwn <. Q , X >. ) )
16 simpl22 1067 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> B e. ( EE ` N ) )
17 simpl23 1068 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> C e. ( EE ` N ) )
18 simpr3r 1050 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , Y >.Cgr <. B , C >. )
19 cgrcom 28033 . . . . . . . . . . . 12 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ Y e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. A , Y >.Cgr <. B , C >. <-> <. B , C >.Cgr <. A , Y >. ) )
203, 5, 10, 16, 17, 19syl122anc 1227 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( <. A , Y >.Cgr <. B , C >. <-> <. B , C >.Cgr <. A , Y >. ) )
2118, 20mpbid 210 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. B , C >.Cgr <. A , Y >. )
22 simpr2r 1048 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. B , C >. )
23 cgrcom 28033 . . . . . . . . . . . 12 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. A , X >.Cgr <. B , C >. <-> <. B , C >.Cgr <. A , X >. ) )
243, 5, 7, 16, 17, 23syl122anc 1227 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( <. A , X >.Cgr <. B , C >. <-> <. B , C >.Cgr <. A , X >. ) )
2522, 24mpbid 210 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. B , C >.Cgr <. A , X >. )
263, 16, 17, 5, 10, 5, 7, 21, 25cgrtr4d 28028 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , Y >.Cgr <. A , X >. )
2715, 6, 26jca32 535 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( A Btwn <. Q , Y >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , Y >.Cgr <. A , X >. ) ) )
28 cgrextend 28051 . . . . . . . 8 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ Y e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( ( A Btwn <. Q , Y >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , Y >.Cgr <. A , X >. ) ) -> <. Q , Y >.Cgr <. Q , X >. ) )
2913, 27, 28sylc 60 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. Q , Y >.Cgr <. Q , X >. )
3029, 26jca 532 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( <. Q , Y >.Cgr <. Q , X >. /\ <. A , Y >.Cgr <. A , X >. ) )
312, 9, 303jca 1168 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( A Btwn <. Q , X >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , X >.Cgr <. A , X >. ) /\ ( <. Q , Y >.Cgr <. Q , X >. /\ <. A , Y >.Cgr <. A , X >. ) ) )
3231ex 434 . . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( ( A Btwn <. Q , X >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , X >.Cgr <. A , X >. ) /\ ( <. Q , Y >.Cgr <. Q , X >. /\ <. A , Y >.Cgr <. A , X >. ) ) ) )
33 simp1 988 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> N e. NN )
34 simp31 1024 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> Q e. ( EE ` N ) )
35 simp21 1021 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
36 simp32 1025 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
37 simp33 1026 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> Y e. ( EE ` N ) )
38 brofs 28048 . . . . 5 |- ( ( ( N e. NN /\ Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. <-> ( ( A Btwn <. Q , X >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , X >.Cgr <. A , X >. ) /\ ( <. Q , Y >.Cgr <. Q , X >. /\ <. A , Y >.Cgr <. A , X >. ) ) ) )
3933, 34, 35, 36, 37, 34, 35, 36, 36, 38syl333anc 1250 . . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. <-> ( ( A Btwn <. Q , X >. /\ A Btwn <. Q , X >. ) /\ ( <. Q , A >.Cgr <. Q , A >. /\ <. A , X >.Cgr <. A , X >. ) /\ ( <. Q , Y >.Cgr <. Q , X >. /\ <. A , Y >.Cgr <. A , X >. ) ) ) )
4032, 39sylibrd 234 . . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. ) )
41 simp1 988 . . . 4 |- ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> Q =/= A )
4241a1i 11 . . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> Q =/= A ) )
4340, 42jcad 533 . 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. /\ Q =/= A ) ) )
44 5segofs 28049 . . 3 |- ( ( ( N e. NN /\ Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. /\ Q =/= A ) -> <. X , Y >.Cgr <. X , X >. ) )
4533, 34, 35, 36, 37, 34, 35, 36, 36, 44syl333anc 1250 . 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( <. <. Q , A >. , <. X , Y >. >. OuterFiveSeg <. <. Q , A >. , <. X , X >. >. /\ Q =/= A ) -> <. X , Y >.Cgr <. X , X >. ) )
46 axcgrid 23174 . . 3 |- ( ( N e. NN /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( <. X , Y >.Cgr <. X , X >. -> X = Y ) )
4733, 36, 37, 36, 46syl13anc 1220 . 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( <. X , Y >.Cgr <. X , X >. -> X = Y ) )
4843, 45, 473syld 55 1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( Q e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( Q =/= A /\ ( A Btwn <. Q , X >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( A Btwn <. Q , Y >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2618   <.cop 3895   class class class wbr 4304   ` cfv 5430   NNcn 10334   EEcee 23146   Btwn cbtwn 23147  Cgrccgr 23148   OuterFiveSeg cofs 28025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-sum 13176  df-ee 23149  df-btwn 23150  df-cgr 23151  df-ofs 28026
This theorem is referenced by:  segconeu  28054  btwnouttr2  28065  cgrxfr  28098  btwnconn1lem2  28131
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