Metamath Proof Explorer

Theorem axeuclid

Description: Euclid's axiom. Take an angle B A C and a point D between B and C . Now, if you extend the segment A D to a point T , then T lies between two points x and y that lie on the angle. Axiom A10 of Schwabhauser p. 13. (Contributed by Scott Fenton, 9-Sep-2013)

Ref Expression
Assertion axeuclid 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , T >. /\ D Btwn <. B , C >. /\ A =/= D ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) ) )

Proof

Step Hyp Ref Expression
1 simpl21 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> A e. ( EE ` N ) )
2 simpl22 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> B e. ( EE ` N ) )
3 1 2 jca 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
4 simpl23 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> C e. ( EE ` N ) )
5 simpl3r 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> T e. ( EE ` N ) )
6 4 5 jca 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) )
7 simprll 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> p e. ( 0 [,] 1 ) )
8 simprlr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> q e. ( 0 [,] 1 ) )
9 simp21 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
10 9 ad2antrr 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) -> A e. ( EE ` N ) )
11 fveecn 
 |-  ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
12 10 11 sylan 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
13 simp3r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> T e. ( EE ` N ) )
14 13 ad2antrr 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) -> T e. ( EE ` N ) )
15 fveecn 
 |-  ( ( T e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( T ` i ) e. CC )
16 14 15 sylan 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( T ` i ) e. CC )
17 mullid 
 |-  ( ( A ` i ) e. CC -> ( 1 x. ( A ` i ) ) = ( A ` i ) )
18 mul02 
 |-  ( ( T ` i ) e. CC -> ( 0 x. ( T ` i ) ) = 0 )
19 17 18 oveqan12d 
 |-  ( ( ( A ` i ) e. CC /\ ( T ` i ) e. CC ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) = ( ( A ` i ) + 0 ) )
20 addrid 
 |-  ( ( A ` i ) e. CC -> ( ( A ` i ) + 0 ) = ( A ` i ) )
21 20 adantr 
 |-  ( ( ( A ` i ) e. CC /\ ( T ` i ) e. CC ) -> ( ( A ` i ) + 0 ) = ( A ` i ) )
22 19 21 eqtrd 
 |-  ( ( ( A ` i ) e. CC /\ ( T ` i ) e. CC ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) = ( A ` i ) )
23 12 16 22 syl2anc 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) = ( A ` i ) )
24 oveq2 
 |-  ( p = 0 -> ( 1 - p ) = ( 1 - 0 ) )
25 1m0e1 
 |-  ( 1 - 0 ) = 1
26 24 25 eqtrdi 
 |-  ( p = 0 -> ( 1 - p ) = 1 )
27 26 oveq1d 
 |-  ( p = 0 -> ( ( 1 - p ) x. ( A ` i ) ) = ( 1 x. ( A ` i ) ) )
28 oveq1 
 |-  ( p = 0 -> ( p x. ( T ` i ) ) = ( 0 x. ( T ` i ) ) )
29 27 28 oveq12d 
 |-  ( p = 0 -> ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) )
30 29 eqeq1d 
 |-  ( p = 0 -> ( ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( A ` i ) <-> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) = ( A ` i ) ) )
31 30 ad2antlr 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( A ` i ) <-> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( T ` i ) ) ) = ( A ` i ) ) )
32 23 31 mpbird 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( A ` i ) )
33 32 eqeq2d 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) <-> ( D ` i ) = ( A ` i ) ) )
34 eqcom 
 |-  ( ( D ` i ) = ( A ` i ) <-> ( A ` i ) = ( D ` i ) )
35 33 34 bitrdi 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) <-> ( A ` i ) = ( D ` i ) ) )
36 35 biimpd 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) -> ( A ` i ) = ( D ` i ) ) )
37 36 adantrd 
 |-  ( ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) /\ i e. ( 1 ... N ) ) -> ( ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) -> ( A ` i ) = ( D ` i ) ) )
38 37 ralimdva 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ p = 0 ) -> ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) -> A. i e. ( 1 ... N ) ( A ` i ) = ( D ` i ) ) )
39 38 impancom 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> ( p = 0 -> A. i e. ( 1 ... N ) ( A ` i ) = ( D ` i ) ) )
40 9 ad2antrr 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> A e. ( EE ` N ) )
41 simp3l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
42 41 ad2antrr 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> D e. ( EE ` N ) )
43 eqeefv 
 |-  ( ( A e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( A = D <-> A. i e. ( 1 ... N ) ( A ` i ) = ( D ` i ) ) )
44 40 42 43 syl2anc 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> ( A = D <-> A. i e. ( 1 ... N ) ( A ` i ) = ( D ` i ) ) )
45 39 44 sylibrd 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> ( p = 0 -> A = D ) )
46 45 necon3d 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) ) -> ( A =/= D -> p =/= 0 ) )
47 46 impr 
 |-  ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) -> p =/= 0 )
48 47 anasss 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> p =/= 0 )
49 eqtr2 
 |-  ( ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) -> ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) )
50 49 ralimi 
 |-  ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) -> A. i e. ( 1 ... N ) ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) )
51 50 adantr 
 |-  ( ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) -> A. i e. ( 1 ... N ) ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) )
52 51 ad2antll 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> A. i e. ( 1 ... N ) ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) )
53 axeuclidlem 
 |-  ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) /\ p =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
54 3 6 7 8 48 52 53 syl231anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
55 54 exp32 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( p e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) -> ( ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) ) ) )
56 55 rexlimdvv 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) ) )
57 brbtwn 
 |-  ( ( D e. ( EE ` N ) /\ A e. ( EE ` N ) /\ T e. ( EE ` N ) ) -> ( D Btwn <. A , T >. <-> E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) ) )
58 41 9 13 57 syl3anc 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( D Btwn <. A , T >. <-> E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) ) )
59 simp22 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
60 simp23 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
61 brbtwn 
 |-  ( ( D e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( D Btwn <. B , C >. <-> E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
62 41 59 60 61 syl3anc 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( D Btwn <. B , C >. <-> E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
63 58 62 3anbi12d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , T >. /\ D Btwn <. B , C >. /\ A =/= D ) <-> ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) /\ A =/= D ) ) )
64 r19.26 
 |-  ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) <-> ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
65 64 2rexbii 
 |-  ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) <-> E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
66 reeanv 
 |-  ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) <-> ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
67 65 66 bitri 
 |-  ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) <-> ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) )
68 67 anbi1i 
 |-  ( ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) <-> ( ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) )
69 r19.41vv 
 |-  ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) <-> ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) )
70 df-3an 
 |-  ( ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) /\ A =/= D ) <-> ( ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) )
71 68 69 70 3bitr4i 
 |-  ( E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) <-> ( E. p e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) /\ A =/= D ) )
72 63 71 bitr4di 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , T >. /\ D Btwn <. B , C >. /\ A =/= D ) <-> E. p e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - p ) x. ( A ` i ) ) + ( p x. ( T ` i ) ) ) /\ ( D ` i ) = ( ( ( 1 - q ) x. ( B ` i ) ) + ( q x. ( C ` i ) ) ) ) /\ A =/= D ) ) )
73 simpl22 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
74 simpl21 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
75 simprl 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> x e. ( EE ` N ) )
76 brbtwn 
 |-  ( ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ x e. ( EE ` N ) ) -> ( B Btwn <. A , x >. <-> E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) ) )
77 73 74 75 76 syl3anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( B Btwn <. A , x >. <-> E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) ) )
78 simpl23 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
79 simprr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> y e. ( EE ` N ) )
80 brbtwn 
 |-  ( ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ y e. ( EE ` N ) ) -> ( C Btwn <. A , y >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) ) )
81 78 74 79 80 syl3anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( C Btwn <. A , y >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) ) )
82 simpl3r 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> T e. ( EE ` N ) )
83 brbtwn 
 |-  ( ( T e. ( EE ` N ) /\ x e. ( EE ` N ) /\ y e. ( EE ` N ) ) -> ( T Btwn <. x , y >. <-> E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
84 82 75 79 83 syl3anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( T Btwn <. x , y >. <-> E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
85 77 81 84 3anbi123d 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) ) )
86 r19.26-3 
 |-  ( A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) <-> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
87 86 rexbii 
 |-  ( E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) <-> E. u e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
88 87 2rexbii 
 |-  ( E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) <-> E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
89 3reeanv 
 |-  ( E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
90 88 89 bitri 
 |-  ( E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )
91 85 90 bitr4di 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( x e. ( EE ` N ) /\ y e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) <-> E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) ) )
92 91 2rexbidva 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( E. x e. ( EE ` N ) E. y e. ( EE ` N ) ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) <-> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) ) )
93 56 72 92 3imtr4d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , T >. /\ D Btwn <. B , C >. /\ A =/= D ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) ) )