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Theorem outsideofeq 32237
Description: Uniqueness law for OutsideOf. Analogue of segconeq 32117. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeq |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
Proof of Theorem outsideofeq
StepHypRef Expression
1 simp1 1061 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> N e. NN )
2 simp21 1094 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3 simp32 1098 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
4 simp22 1095 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> R e. ( EE ` N ) )
5 broutsideof2 32229 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
61, 2, 3, 4, 5syl13anc 1328 . . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
76anbi1d 741 . . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) <-> ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) ) )
8 simp33 1099 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> Y e. ( EE ` N ) )
9 broutsideof2 32229 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
101, 2, 8, 4, 9syl13anc 1328 . . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
1110anbi1d 741 . . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) <-> ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) )
127, 11anbi12d 747 . 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) <-> ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) )
13 simpll3 1102 . . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) )
14 simprl3 1108 . . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) )
1513, 14jca 554 . . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
1615adantl 482 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
17 simpll2 1101 . . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> R =/= A )
1817adantl 482 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> R =/= A )
19 simp23 1096 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
20 simp31 1097 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
21 simprlr 803 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. B , C >. )
22 simprrr 805 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , Y >.Cgr <. B , C >. )
231, 2, 3, 2, 8, 19, 20, 21, 22cgrtr3and 32102 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
2416, 18, 23jca32 558 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) )
25 simprll 802 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
26 simprlr 803 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
27 simprrr 805 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
28 midofsegid 32211 . . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
291, 2, 4, 3, 8, 28syl122anc 1335 . . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
3029adantr 481 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
3125, 26, 27, 30mp3and 1427 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
3231exp32 631 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
33 simprlr 803 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
34 simprll 802 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
351, 2, 8, 4, 3, 33, 34btwnexchand 32133 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , X >. )
36 simprrr 805 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
371, 2, 3, 8, 35, 36endofsegidand 32193 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
3837exp32 631 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
39 simprll 802 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
40 simprlr 803 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
411, 2, 3, 4, 8, 39, 40btwnexchand 32133 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , Y >. )
42 simprrr 805 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
431, 2, 3, 2, 8, 42cgrcomand 32098 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , Y >.Cgr <. A , X >. )
441, 2, 8, 3, 41, 43endofsegidand 32193 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y = X )
4544eqcomd 2628 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
4645exp32 631 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
47 simprr 796 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X Btwn <. A , Y >. )
48 simplrr 801 . . . . . . . . . . . . 13 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) -> <. A , X >.Cgr <. A , Y >. )
4948adantl 482 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , X >.Cgr <. A , Y >. )
501, 2, 3, 2, 8, 49cgrcomand 32098 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , Y >.Cgr <. A , X >. )
511, 2, 8, 3, 47, 50endofsegidand 32193 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> Y = X )
5251eqcomd 2628 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X = Y )
5352expr 643 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. -> X = Y ) )
54 simprr 796 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> Y Btwn <. A , X >. )
55 simplrr 801 . . . . . . . . . . 11 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) -> <. A , X >.Cgr <. A , Y >. )
5655adantl 482 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> <. A , X >.Cgr <. A , Y >. )
571, 2, 3, 8, 54, 56endofsegidand 32193 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> X = Y )
5857expr 643 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( Y Btwn <. A , X >. -> X = Y ) )
59 simprrl 804 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R =/= A )
6059necomd 2849 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> A =/= R )
61 simprll 802 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
62 simprlr 803 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
63 btwnconn1 32208 . . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
641, 2, 4, 3, 8, 63syl122anc 1335 . . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
6564adantr 481 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
6660, 61, 62, 65mp3and 1427 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) )
6753, 58, 66mpjaod 396 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
6867exp32 631 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
6932, 38, 46, 68ccased 988 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
7069imp32 449 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
7124, 70syldan 487 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> X = Y )
7271ex 450 . 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
7312, 72sylbid 230 1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ 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  OutsideOfcoutsideof 32226
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  df-colinear 32146  df-ifs 32147  df-cgr3 32148  df-fs 32149  df-outsideof 32227
This theorem is referenced by:  outsideofeu  32238  outsidele  32239
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