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Theorem outsideofeu 32238
Description: Given a non-degenerate ray, there is a unique point congruent to the segment B C lying on the ray A R. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeu |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( R =/= A /\ B =/= C ) -> E! x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) ) )
Distinct variable groups:   x, A   x, B   x, C   x, N   x, R
Proof of Theorem outsideofeu
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 segcon2 32212 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) /\ <. A , x >.Cgr <. B , C >. ) )
21adantr 481 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> E. x e. ( EE ` N ) ( ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) /\ <. A , x >.Cgr <. B , C >. ) )
3 simpl1 1064 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> N e. NN )
4 simpl2l 1114 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> A e. ( EE ` N ) )
5 simpr 477 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> x e. ( EE ` N ) )
6 simpl2r 1115 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> R e. ( EE ` N ) )
7 broutsideof2 32229 . . . . . . . . . . . 12 |- ( ( 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 >. ) ) ) )
83, 4, 5, 6, 7syl13anc 1328 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( AOutsideOf <. x , R >. <-> ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) )
98adantr 481 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) ) -> ( AOutsideOf <. x , R >. <-> ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) )
10 simp3 1063 . . . . . . . . . . 11 |- ( ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) -> ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) )
11 simpllr 799 . . . . . . . . . . . . . . . 16 |- ( ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) -> B =/= C )
1211adantl 482 . . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> B =/= C )
13 simprlr 803 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> <. A , x >.Cgr <. B , C >. )
14 simp2l 1087 . . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
1514anim1i 592 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( A e. ( EE ` N ) /\ x e. ( EE ` N ) ) )
16 simpl3 1066 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
17 cgrdegen 32111 . . . . . . . . . . . . . . . . . . 19 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ x e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. A , x >.Cgr <. B , C >. -> ( A = x <-> B = C ) ) )
183, 15, 16, 17syl3anc 1326 . . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( <. A , x >.Cgr <. B , C >. -> ( A = x <-> B = C ) ) )
1918adantr 481 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> ( <. A , x >.Cgr <. B , C >. -> ( A = x <-> B = C ) ) )
2013, 19mpd 15 . . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> ( A = x <-> B = C ) )
2120necon3bid 2838 . . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> ( A =/= x <-> B =/= C ) )
2212, 21mpbird 247 . . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> A =/= x )
2322necomd 2849 . . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> x =/= A )
24 simplll 798 . . . . . . . . . . . . . 14 |- ( ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) -> R =/= A )
2524adantl 482 . . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> R =/= A )
26 simprr 796 . . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) )
2723, 25, 263jca 1242 . . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) -> ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) )
2827expr 643 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) ) -> ( ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) -> ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) ) )
2910, 28impbid2 216 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) ) -> ( ( x =/= A /\ R =/= A /\ ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) <-> ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) )
309, 29bitrd 268 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) ) -> ( AOutsideOf <. x , R >. <-> ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) ) )
31 orcom 402 . . . . . . . . 9 |- ( ( x Btwn <. A , R >. \/ R Btwn <. A , x >. ) <-> ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) )
3230, 31syl6bb 276 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( R =/= A /\ B =/= C ) /\ <. A , x >.Cgr <. B , C >. ) ) -> ( AOutsideOf <. x , R >. <-> ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) ) )
3332expr 643 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( R =/= A /\ B =/= C ) ) -> ( <. A , x >.Cgr <. B , C >. -> ( AOutsideOf <. x , R >. <-> ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) ) ) )
3433pm5.32rd 672 . . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( R =/= A /\ B =/= C ) ) -> ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) <-> ( ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) /\ <. A , x >.Cgr <. B , C >. ) ) )
3534an32s 846 . . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) /\ x e. ( EE ` N ) ) -> ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) <-> ( ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) /\ <. A , x >.Cgr <. B , C >. ) ) )
3635rexbidva 3049 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> ( E. x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) <-> E. x e. ( EE ` N ) ( ( R Btwn <. A , x >. \/ x Btwn <. A , R >. ) /\ <. A , x >.Cgr <. B , C >. ) ) )
372, 36mpbird 247 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> E. x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) )
38 simpl1 1064 . . . . . . . 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 ) ) ) -> N e. NN )
39 simpl2l 1114 . . . . . . . . 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 ) ) ) -> A e. ( EE ` N ) )
40 simpl2r 1115 . . . . . . . . 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 e. ( EE ` N ) )
41 simpl3l 1116 . . . . . . . . 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 ) ) ) -> B e. ( EE ` N ) )
4239, 40, 413jca 1242 . . . . . . . 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 ) ) ) -> ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
43 simpl3r 1117 . . . . . . . . 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 ) ) ) -> C e. ( EE ` N ) )
44 simprl 794 . . . . . . . . 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 e. ( EE ` N ) )
45 simprr 796 . . . . . . . . 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 ) ) ) -> y e. ( EE ` N ) )
4643, 44, 453jca 1242 . . . . . . . 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 ) ) ) -> ( C e. ( EE ` N ) /\ x e. ( EE ` N ) /\ y e. ( EE ` N ) ) )
4738, 42, 463jca 1242 . . . . . . 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 ) ) ) -> ( 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 ) ) ) )
48 simpr 477 . . . . . . 7 |- ( ( ( R =/= A /\ B =/= C ) /\ ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) ) -> ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) )
49 outsideofeq 32237 . . . . . . . 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 ) ) ) -> ( ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) -> x = y ) )
5049imp 445 . . . . . . 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 ) ) ) /\ ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) ) -> x = y )
5147, 48, 50syl2an 494 . . . . . 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 =/= A /\ B =/= C ) /\ ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) ) ) -> x = y )
5251an4s 869 . . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) /\ ( ( 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 )
5352exp32 631 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> ( ( 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 ) ) )
5453ralrimivv 2970 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> A. x e. ( EE ` N ) A. y e. ( EE ` N ) ( ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) -> x = y ) )
55 opeq1 4402 . . . . . 6 |- ( x = y -> <. x , R >. = <. y , R >. )
5655breq2d 4665 . . . . 5 |- ( x = y -> ( AOutsideOf <. x , R >. <-> AOutsideOf <. y , R >. ) )
57 opeq2 4403 . . . . . 6 |- ( x = y -> <. A , x >. = <. A , y >. )
5857breq1d 4663 . . . . 5 |- ( x = y -> ( <. A , x >.Cgr <. B , C >. <-> <. A , y >.Cgr <. B , C >. ) )
5956, 58anbi12d 747 . . . 4 |- ( x = y -> ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) <-> ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) )
6059reu4 3400 . . 3 |- ( E! x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) <-> ( E. x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ A. x e. ( EE ` N ) A. y e. ( EE ` N ) ( ( ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) /\ ( AOutsideOf <. y , R >. /\ <. A , y >.Cgr <. B , C >. ) ) -> x = y ) ) )
6137, 54, 60sylanbrc 698 . 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( R =/= A /\ B =/= C ) ) -> E! x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) )
6261ex 450 1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( R =/= A /\ B =/= C ) -> E! x e. ( EE ` N ) ( AOutsideOf <. x , R >. /\ <. A , x >.Cgr <. B , C >. ) ) )
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   A.wral 2912   E.wrex 2913   E!wreu 2914   <.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: (None)
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