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Theorem segconeq 32117
Description: Two points that satsify the conclusion of axsegcon 25807 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 1120 . . . . . . 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 554 . . . . . 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 1064 . . . . . . . 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 1142 . . . . . . . 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 1139 . . . . . . . 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 32095 . . . . . . 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 1143 . . . . . . . 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 32095 . . . . . . 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 554 . . . . . 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 1144 . . . . . . . . . 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 1242 . . . . . . . . 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 1242 . . . . . . . . 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 1242 . . . . . . . 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 1122 . . . . . . . . . 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 554 . . . . . . . . 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 1140 . . . . . . . . . 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 1141 . . . . . . . . . 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 1123 . . . . . . . . . . 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 32097 . . . . . . . . . . . 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 1335 . . . . . . . . . . 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 222 . . . . . . . . . 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 1121 . . . . . . . . . . 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 32097 . . . . . . . . . . . 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 1335 . . . . . . . . . . 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 222 . . . . . . . . . 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 32092 . . . . . . . . 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 558 . . . . . . . 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 32115 . . . . . . . 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 65 . . . . . . 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 554 . . . . . 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 1242 . . . . 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 450 . . . 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 1061 . . . . 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 1097 . . . . 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 1094 . . . . 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 1098 . . . . 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 1099 . . . . 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 32112 . . . . 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 1358 . . . 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 249 . . 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 1061 . . . 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 555 . 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 32113 . . 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 1358 . 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 25796 . . 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 1328 . 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 60 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   OuterFiveSeg cofs 32089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090
This theorem is referenced by:  segconeu  32118  btwnouttr2  32129  cgrxfr  32162  btwnconn1lem2  32195
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