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Theorem axeuclid 24468
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.)
Assertion
Ref Expression
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 >.
) ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, D, y   
x, N, y    x, T, y

Proof of Theorem axeuclid
Dummy variables  i  p  q  r  s  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl21 1072 . . . . . 6  |-  ( ( ( 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 1073 . . . . . 6  |-  ( ( ( 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 ) )
31, 2jca 530 . . . . 5  |-  ( ( ( 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 1074 . . . . . 6  |-  ( ( ( 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 1050 . . . . . 6  |-  ( ( ( 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 ) )
64, 5jca 530 . . . . 5  |-  ( ( ( 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 761 . . . . 5  |-  ( ( ( 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 762 . . . . 5  |-  ( ( ( 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 1027 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 ) )
109ad2antrr 723 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 24407 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
1210, 11sylan 469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( 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 1023 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 ) )
1413ad2antrr 723 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 24407 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( T `  i )  e.  CC )
1614, 15sylan 469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( 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 mulid2 9583 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
18 mul02 9747 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T `  i )  e.  CC  ->  (
0  x.  ( T `
 i ) )  =  0 )
1917, 18oveqan12d 6289 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A `  i
)  e.  CC  /\  ( T `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( T `
 i ) ) )  =  ( ( A `  i )  +  0 ) )
20 addid1 9749 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A `  i )  e.  CC  ->  (
( A `  i
)  +  0 )  =  ( A `  i ) )
2120adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A `  i
)  e.  CC  /\  ( T `  i )  e.  CC )  -> 
( ( A `  i )  +  0 )  =  ( A `
 i ) )
2219, 21eqtrd 2495 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A `  i
)  e.  CC  /\  ( T `  i )  e.  CC )  -> 
( ( 1  x.  ( A `  i
) )  +  ( 0  x.  ( T `
 i ) ) )  =  ( A `
 i ) )
2312, 16, 22syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 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 6278 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p  =  0  ->  (
1  -  p )  =  ( 1  -  0 ) )
25 1m0e1 10642 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  -  0 )  =  1
2624, 25syl6eq 2511 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  =  0  ->  (
1  -  p )  =  1 )
2726oveq1d 6285 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  ( A `
 i ) )  =  ( 1  x.  ( A `  i
) ) )
28 oveq1 6277 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  0  ->  (
p  x.  ( T `
 i ) )  =  ( 0  x.  ( T `  i
) ) )
2927, 28oveq12d 6288 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  0  ->  (
( ( 1  -  p )  x.  ( A `  i )
)  +  ( p  x.  ( T `  i ) ) )  =  ( ( 1  x.  ( A `  i ) )  +  ( 0  x.  ( T `  i )
) ) )
3029eqeq1d 2456 . . . . . . . . . . . . . . . . 17  |-  ( p  =  0  ->  (
( ( ( 1  -  p )  x.  ( A `  i
) )  +  ( p  x.  ( T `
 i ) ) )  =  ( A `
 i )  <->  ( (
1  x.  ( A `
 i ) )  +  ( 0  x.  ( T `  i
) ) )  =  ( A `  i
) ) )
3130ad2antlr 724 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 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 ) ) )
3223, 31mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( 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 ) )
3332eqeq2d 2468 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 2463 . . . . . . . . . . . . . 14  |-  ( ( D `  i )  =  ( A `  i )  <->  ( A `  i )  =  ( D `  i ) )
3533, 34syl6bb 261 . . . . . . . . . . . . 13  |-  ( ( ( ( ( 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 ) ) )
3635biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 ) ) )
3736adantrd 466 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) ) )
3837ralimdva 2862 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )
3938impancom 438 . . . . . . . . 9  |-  ( ( ( ( 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 ) ) )
409ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( 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 1022 . . . . . . . . . . 11  |-  ( ( 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 ) )
4241ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( 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 24408 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( A  =  D  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( D `  i ) ) )
4440, 42, 43syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( 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 ) ) )
4539, 44sylibrd 234 . . . . . . . 8  |-  ( ( ( ( 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 ) )
4645necon3d 2678 . . . . . . 7  |-  ( ( ( ( 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 ) )
4746impr 617 . . . . . 6  |-  ( ( ( ( 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
)
4847anasss 645 . . . . 5  |-  ( ( ( 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 2481 . . . . . . . 8  |-  ( ( ( 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
) ) ) )
5049ralimi 2847 . . . . . . 7  |-  ( 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 )
) ) )
5150adantr 463 . . . . . 6  |-  ( ( 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 ) ) ) )
5251ad2antll 726 . . . . 5  |-  ( ( ( 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 24467 . . . . 5  |-  ( ( ( ( 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 ) ) ) ) )
543, 6, 7, 8, 48, 52, 53syl231anc 1246 . . . 4  |-  ( ( ( 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 ) ) ) ) )
5554exp32 603 . . 3  |-  ( ( 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 ) ) ) ) ) ) )
5655rexlimdvv 2952 . 2  |-  ( ( 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 24404 . . . . 5  |-  ( ( 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
) ) ) ) )
5841, 9, 13, 57syl3anc 1226 . . . 4  |-  ( ( 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 1028 . . . . 5  |-  ( ( 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 1029 . . . . 5  |-  ( ( 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 24404 . . . . 5  |-  ( ( 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
) ) ) ) )
6241, 59, 60, 61syl3anc 1226 . . . 4  |-  ( ( 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
) ) ) ) )
6358, 623anbi12d 1298 . . 3  |-  ( ( 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 2981 . . . . . . 7  |-  ( 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 )
) ) ) )
65642rexbii 2957 . . . . . 6  |-  ( 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 3022 . . . . . 6  |-  ( 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 )
) ) ) )
6765, 66bitri 249 . . . . 5  |-  ( 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 )
) ) ) )
6867anbi1i 693 . . . 4  |-  ( ( 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 3008 . . . 4  |-  ( 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 973 . . . 4  |-  ( ( 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 ) )
7168, 69, 703bitr4i 277 . . 3  |-  ( 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 ) )
7263, 71syl6bbr 263 . 2  |-  ( ( 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 1073 . . . . . 6  |-  ( ( ( 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 1072 . . . . . 6  |-  ( ( ( 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 754 . . . . . 6  |-  ( ( ( 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 24404 . . . . . 6  |-  ( ( 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
) ) ) ) )
7773, 74, 75, 76syl3anc 1226 . . . . 5  |-  ( ( ( 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 1074 . . . . . 6  |-  ( ( ( 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 755 . . . . . 6  |-  ( ( ( 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 24404 . . . . . 6  |-  ( ( 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
) ) ) ) )
8178, 74, 79, 80syl3anc 1226 . . . . 5  |-  ( ( ( 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 1050 . . . . . 6  |-  ( ( ( 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 24404 . . . . . 6  |-  ( ( 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
) ) ) ) )
8482, 75, 79, 83syl3anc 1226 . . . . 5  |-  ( ( ( 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 )
) ) ) )
8577, 81, 843anbi123d 1297 . . . 4  |-  ( ( ( 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 2983 . . . . . . 7  |-  ( 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 )
) ) ) )
8786rexbii 2956 . . . . . 6  |-  ( 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 ) ) ) ) )
88872rexbii 2957 . . . . 5  |-  ( 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 3023 . . . . 5  |-  ( 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 )
) ) ) )
9088, 89bitri 249 . . . 4  |-  ( 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 )
) ) ) )
9185, 90syl6bbr 263 . . 3  |-  ( ( ( 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 ) ) ) ) ) )
92912rexbidva 2971 . 2  |-  ( ( 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 ) ) ) ) ) )
9356, 72, 923imtr4d 268 1  |-  ( ( 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 >.
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   NNcn 10531   [,]cicc 11535   ...cfz 11675   EEcee 24393    Btwn cbtwn 24394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-z 10861  df-uz 11083  df-icc 11539  df-fz 11676  df-ee 24396  df-btwn 24397
This theorem is referenced by:  eengtrkge  24491
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