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Theorem ax5seglem6 25838
Description: Lemma for ax5seg 25842. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
ax5seglem6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  T  =  S )
Distinct variable groups:    A, i    B, i    C, i    D, i   
i, E    i, F    i, N    S, i    T, i

Proof of Theorem ax5seglem6
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 simp22l 1076 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  T  e.  ( 0 [,] 1
) )
2 0re 9081 . . . . . 6  |-  0  e.  RR
3 1re 9080 . . . . . 6  |-  1  e.  RR
42, 3elicc2i 10966 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
54simp1bi 972 . . . 4  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
6 resqcl 11439 . . . . 5  |-  ( T  e.  RR  ->  ( T ^ 2 )  e.  RR )
76recnd 9104 . . . 4  |-  ( T  e.  RR  ->  ( T ^ 2 )  e.  CC )
81, 5, 73syl 19 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( T ^ 2 )  e.  CC )
9 simp22r 1077 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  S  e.  ( 0 [,] 1
) )
102, 3elicc2i 10966 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1110simp1bi 972 . . . 4  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
12 resqcl 11439 . . . . 5  |-  ( S  e.  RR  ->  ( S ^ 2 )  e.  RR )
1312recnd 9104 . . . 4  |-  ( S  e.  RR  ->  ( S ^ 2 )  e.  CC )
149, 11, 133syl 19 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( S ^ 2 )  e.  CC )
15 fzfid 11302 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( 1 ... N )  e. 
Fin )
16 simprl1 1002 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  A  e.  ( EE `  N ) )
17163ad2ant1 978 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  A  e.  ( EE `  N ) )
18 fveecn 25806 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( A `  j )  e.  CC )
1917, 18sylan 458 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  /\  j  e.  ( 1 ... N
) )  ->  ( A `  j )  e.  CC )
20 simprl3 1004 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  C  e.  ( EE `  N ) )
21203ad2ant1 978 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  C  e.  ( EE `  N ) )
22 fveecn 25806 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( C `  j )  e.  CC )
2321, 22sylan 458 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  /\  j  e.  ( 1 ... N
) )  ->  ( C `  j )  e.  CC )
2419, 23subcld 9401 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( A `  j
)  -  ( C `
 j ) )  e.  CC )
2524sqcld 11511 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( ( A `  j )  -  ( C `  j )
) ^ 2 )  e.  CC )
2615, 25fsumcl 12517 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  e.  CC )
27 simp1l 981 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  N  e.  NN )
28 simp1rl 1022 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
29 simp21 990 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  A  =/=  B )
30 simp23l 1078 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) )
31 ax5seglem5 25837 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  T  e.  ( 0 [,] 1
)  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =/=  0
)
3227, 28, 29, 1, 30, 31syl23anc 1191 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =/=  0
)
33 simp3l 985 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  <. A ,  B >.Cgr <. D ,  E >. )
34 simprl2 1003 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  B  e.  ( EE `  N ) )
35 simprr1 1005 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  D  e.  ( EE `  N ) )
36 simprr2 1006 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  E  e.  ( EE `  N ) )
37 brcgr 25804 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  <->  sum_ j  e.  ( 1 ... N ) ( ( ( A `  j )  -  ( B `  j )
) ^ 2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `
 j )  -  ( E `  j ) ) ^ 2 ) ) )
3816, 34, 35, 36, 37syl22anc 1185 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  <->  sum_ j  e.  ( 1 ... N ) ( ( ( A `  j )  -  ( B `  j )
) ^ 2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `
 j )  -  ( E `  j ) ) ^ 2 ) ) )
39383ad2ant1 978 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >. 
<-> 
sum_ j  e.  ( 1 ... N ) ( ( ( A `
 j )  -  ( B `  j ) ) ^ 2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `
 j )  -  ( E `  j ) ) ^ 2 ) ) )
4033, 39mpbid 202 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( B `  j ) ) ^
2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `  j )  -  ( E `  j )
) ^ 2 ) )
41 ax5seglem1 25832 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( T  e.  (
0 [,] 1 )  /\  A. i  e.  ( 1 ... N
) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i )
)  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( B `  j ) ) ^
2 )  =  ( ( T ^ 2 )  x.  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 ) ) )
4227, 17, 21, 1, 30, 41syl122anc 1193 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( B `  j ) ) ^
2 )  =  ( ( T ^ 2 )  x.  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 ) ) )
43353ad2ant1 978 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  D  e.  ( EE `  N ) )
44 simprr3 1007 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  ->  F  e.  ( EE `  N ) )
45443ad2ant1 978 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  F  e.  ( EE `  N ) )
46 simp23r 1079 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  A. i  e.  ( 1 ... N
) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i )
)  +  ( S  x.  ( F `  i ) ) ) )
47 ax5seglem1 25832 . . . . . 6  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) )  /\  ( S  e.  (
0 [,] 1 )  /\  A. i  e.  ( 1 ... N
) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i )
)  +  ( S  x.  ( F `  i ) ) ) ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( D `  j )  -  ( E `  j ) ) ^
2 )  =  ( ( S ^ 2 )  x.  sum_ j  e.  ( 1 ... N
) ( ( ( D `  j )  -  ( F `  j ) ) ^
2 ) ) )
4827, 43, 45, 9, 46, 47syl122anc 1193 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( D `  j )  -  ( E `  j ) ) ^
2 )  =  ( ( S ^ 2 )  x.  sum_ j  e.  ( 1 ... N
) ( ( ( D `  j )  -  ( F `  j ) ) ^
2 ) ) )
4940, 42, 483eqtr3d 2475 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( ( T ^ 2 )  x. 
sum_ j  e.  ( 1 ... N ) ( ( ( A `
 j )  -  ( C `  j ) ) ^ 2 ) )  =  ( ( S ^ 2 )  x.  sum_ j  e.  ( 1 ... N ) ( ( ( D `
 j )  -  ( F `  j ) ) ^ 2 ) ) )
50 simp1rr 1023 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( D  e.  ( EE `  N
)  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )
51 simp22 991 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( T  e.  ( 0 [,] 1
)  /\  S  e.  ( 0 [,] 1
) ) )
52 simp23 992 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )
53 simp3r 986 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  <. B ,  C >.Cgr <. E ,  F >. )
54 ax5seglem3 25835 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `  j )  -  ( F `  j )
) ^ 2 ) )
5527, 28, 50, 51, 52, 33, 53, 54syl322anc 1212 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N
) ( ( ( A `  j )  -  ( C `  j ) ) ^
2 )  =  sum_ j  e.  ( 1 ... N ) ( ( ( D `  j )  -  ( F `  j )
) ^ 2 ) )
5655oveq2d 6089 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( ( S ^ 2 )  x. 
sum_ j  e.  ( 1 ... N ) ( ( ( A `
 j )  -  ( C `  j ) ) ^ 2 ) )  =  ( ( S ^ 2 )  x.  sum_ j  e.  ( 1 ... N ) ( ( ( D `
 j )  -  ( F `  j ) ) ^ 2 ) ) )
5749, 56eqtr4d 2470 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( ( T ^ 2 )  x. 
sum_ j  e.  ( 1 ... N ) ( ( ( A `
 j )  -  ( C `  j ) ) ^ 2 ) )  =  ( ( S ^ 2 )  x.  sum_ j  e.  ( 1 ... N ) ( ( ( A `
 j )  -  ( C `  j ) ) ^ 2 ) ) )
588, 14, 26, 32, 57mulcan2ad 9648 . 2  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( T ^ 2 )  =  ( S ^ 2 ) )
594simp2bi 973 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
605, 59jca 519 . . . 4  |-  ( T  e.  ( 0 [,] 1 )  ->  ( T  e.  RR  /\  0  <_  T ) )
611, 60syl 16 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( T  e.  RR  /\  0  <_  T ) )
6210simp2bi 973 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
6311, 62jca 519 . . . 4  |-  ( S  e.  ( 0 [,] 1 )  ->  ( S  e.  RR  /\  0  <_  S ) )
649, 63syl 16 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( S  e.  RR  /\  0  <_  S ) )
65 sq11 11444 . . 3  |-  ( ( ( T  e.  RR  /\  0  <_  T )  /\  ( S  e.  RR  /\  0  <_  S )
)  ->  ( ( T ^ 2 )  =  ( S ^ 2 )  <->  T  =  S
) )
6661, 64, 65syl2anc 643 . 2  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  ( ( T ^ 2 )  =  ( S ^ 2 )  <->  T  =  S
) )
6758, 66mpbid 202 1  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  /\  ( A. i  e.  ( 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i )
) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  ( ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i )
) ) ) )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) )  ->  T  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985    <_ cle 9111    - cmin 9281   NNcn 9990   2c2 10039   [,]cicc 10909   ...cfz 11033   ^cexp 11372   sum_csu 12469   EEcee 25792  Cgrccgr 25794
This theorem is referenced by:  ax5seg  25842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-ee 25795  df-cgr 25797
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