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Theorem trisegint 32135
Description: A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)
Assertion
Ref Expression
trisegint |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) ) )
Distinct variable groups:   A, q   B, q   C, q   D, q   E, q   N, q   P, q
Proof of Theorem trisegint
Dummy variable r is distinct from all other variables.
StepHypRef Expression
1 simpl1 1064 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> N e. NN )
2 simpl23 1141 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> C e. ( EE ` N ) )
3 simpl21 1139 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> A e. ( EE ` N ) )
4 simpl31 1142 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> D e. ( EE ` N ) )
52, 3, 43jca 1242 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) )
6 simpl32 1143 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> E e. ( EE ` N ) )
7 simpl33 1144 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> P e. ( EE ` N ) )
86, 7jca 554 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( E e. ( EE ` N ) /\ P e. ( EE ` N ) ) )
91, 5, 83jca 1242 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) )
10 simpr2 1068 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> E Btwn <. D , C >. )
11 btwncom 32121 . . . . . . 7 |- ( ( N e. NN /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( E Btwn <. D , C >. <-> E Btwn <. C , D >. ) )
121, 6, 4, 2, 11syl13anc 1328 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( E Btwn <. D , C >. <-> E Btwn <. C , D >. ) )
1310, 12mpbid 222 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> E Btwn <. C , D >. )
14 simpr3 1069 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> P Btwn <. A , D >. )
1513, 14jca 554 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( E Btwn <. C , D >. /\ P Btwn <. A , D >. ) )
16 axpasch 25821 . . . 4 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( ( E Btwn <. C , D >. /\ P Btwn <. A , D >. ) -> E. r e. ( EE ` N ) ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) )
179, 15, 16sylc 65 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> E. r e. ( EE ` N ) ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) )
18 simp1l1 1154 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> N e. NN )
1963ad2ant1 1082 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> E e. ( EE ` N ) )
2023ad2ant1 1082 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> C e. ( EE ` N ) )
2133ad2ant1 1082 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> A e. ( EE ` N ) )
2219, 20, 213jca 1242 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( E e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
23 simp2 1062 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> r e. ( EE ` N ) )
24 simpl22 1140 . . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> B e. ( EE ` N ) )
25243ad2ant1 1082 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> B e. ( EE ` N ) )
2623, 25jca 554 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( r e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
2718, 22, 263jca 1242 . . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( N e. NN /\ ( E e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( r e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) )
28 simp3l 1089 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> r Btwn <. E , A >. )
29 simp1r1 1157 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> B Btwn <. A , C >. )
30 btwncom 32121 . . . . . . . . 9 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. <-> B Btwn <. C , A >. ) )
3118, 25, 21, 20, 30syl13anc 1328 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( B Btwn <. A , C >. <-> B Btwn <. C , A >. ) )
3229, 31mpbid 222 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> B Btwn <. C , A >. )
3328, 32jca 554 . . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( r Btwn <. E , A >. /\ B Btwn <. C , A >. ) )
34 axpasch 25821 . . . . . 6 |- ( ( N e. NN /\ ( E e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( r e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( r Btwn <. E , A >. /\ B Btwn <. C , A >. ) -> E. q e. ( EE ` N ) ( q Btwn <. r , C >. /\ q Btwn <. B , E >. ) ) )
3527, 33, 34sylc 65 . . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> E. q e. ( EE ` N ) ( q Btwn <. r , C >. /\ q Btwn <. B , E >. ) )
36 simpll1 1100 . . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) )
3736, 1syl 17 . . . . . . . . . 10 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> N e. NN )
3836, 7syl 17 . . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> P e. ( EE ` N ) )
39 simpll2 1101 . . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> r e. ( EE ` N ) )
4038, 39jca 554 . . . . . . . . . 10 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> ( P e. ( EE ` N ) /\ r e. ( EE ` N ) ) )
41 simplr 792 . . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> q e. ( EE ` N ) )
4236, 2syl 17 . . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> C e. ( EE ` N ) )
4341, 42jca 554 . . . . . . . . . 10 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> ( q e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
4437, 40, 433jca 1242 . . . . . . . . 9 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> ( N e. NN /\ ( P e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( q e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) )
45 simpl3r 1117 . . . . . . . . . 10 |- ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) -> r Btwn <. P , C >. )
4645anim1i 592 . . . . . . . . 9 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> ( r Btwn <. P , C >. /\ q Btwn <. r , C >. ) )
47 btwnexch2 32130 . . . . . . . . 9 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( q e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( r Btwn <. P , C >. /\ q Btwn <. r , C >. ) -> q Btwn <. P , C >. ) )
4844, 46, 47sylc 65 . . . . . . . 8 |- ( ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) /\ q Btwn <. r , C >. ) -> q Btwn <. P , C >. )
4948ex 450 . . . . . . 7 |- ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) -> ( q Btwn <. r , C >. -> q Btwn <. P , C >. ) )
5049anim1d 588 . . . . . 6 |- ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) /\ q e. ( EE ` N ) ) -> ( ( q Btwn <. r , C >. /\ q Btwn <. B , E >. ) -> ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) ) )
5150reximdva 3017 . . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> ( E. q e. ( EE ` N ) ( q Btwn <. r , C >. /\ q Btwn <. B , E >. ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) ) )
5235, 51mpd 15 . . . 4 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) /\ r e. ( EE ` N ) /\ ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) )
5352rexlimdv3a 3033 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> ( E. r e. ( EE ` N ) ( r Btwn <. E , A >. /\ r Btwn <. P , C >. ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) ) )
5417, 53mpd 15 . 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) )
5554ex 450 1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ E Btwn <. D , C >. /\ P Btwn <. A , D >. ) -> E. q e. ( EE ` N ) ( q Btwn <. P , C >. /\ q Btwn <. B , E >. ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769
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: (None)
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