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Theorem ax5seglem6 24654
Description: Lemma for ax5seg 24658. 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 1116 . . . 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 9626 . . . . . 6  |-  0  e.  RR
3 1re 9625 . . . . . 6  |-  1  e.  RR
42, 3elicc2i 11644 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
54simp1bi 1012 . . . 4  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
6 resqcl 12280 . . . . 5  |-  ( T  e.  RR  ->  ( T ^ 2 )  e.  RR )
76recnd 9652 . . . 4  |-  ( T  e.  RR  ->  ( T ^ 2 )  e.  CC )
81, 5, 73syl 18 . . 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 1117 . . . 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 11644 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1110simp1bi 1012 . . . 4  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
12 resqcl 12280 . . . . 5  |-  ( S  e.  RR  ->  ( S ^ 2 )  e.  RR )
1312recnd 9652 . . . 4  |-  ( S  e.  RR  ->  ( S ^ 2 )  e.  CC )
149, 11, 133syl 18 . . 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 12124 . . . 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 1042 . . . . . . . 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 1018 . . . . . . 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 24622 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( A `  j )  e.  CC )
1917, 18sylan 469 . . . . . 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 1044 . . . . . . . 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 1018 . . . . . . 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 24622 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( C `  j )  e.  CC )
2321, 22sylan 469 . . . . . 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 9967 . . . . 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 12352 . . . 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 13704 . . 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 1021 . . . 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 1062 . . . 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 1030 . . . 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 1118 . . . 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 24653 . . . 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 1237 . . 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 1025 . . . . . 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 1043 . . . . . . . 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 1045 . . . . . . . 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 1046 . . . . . . . 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 24620 . . . . . . . 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 1231 . . . . . . 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 1018 . . . . . 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 210 . . . . 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 24648 . . . . . 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 1239 . . . . 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 1018 . . . . . 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 1047 . . . . . . 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 1018 . . . . . 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 1119 . . . . . 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 24648 . . . . . 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 1239 . . . . 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 2451 . . . 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 1063 . . . . . 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 1031 . . . . . 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 1032 . . . . . 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 1026 . . . . . 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 24651 . . . . . 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 1258 . . . . 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 6294 . . . 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 2446 . . 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 10226 . 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 1013 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
605, 59jca 530 . . . 4  |-  ( T  e.  ( 0 [,] 1 )  ->  ( T  e.  RR  /\  0  <_  T ) )
611, 60syl 17 . . 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 1013 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
6311, 62jca 530 . . . 4  |-  ( S  e.  ( 0 [,] 1 )  ->  ( S  e.  RR  /\  0  <_  S ) )
649, 63syl 17 . . 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 12285 . . 3  |-  ( ( ( T  e.  RR  /\  0  <_  T )  /\  ( S  e.  RR  /\  0  <_  S )
)  ->  ( ( T ^ 2 )  =  ( S ^ 2 )  <->  T  =  S
) )
6661, 64, 65syl2anc 659 . 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 210 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   <.cop 3978   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    <_ cle 9659    - cmin 9841   NNcn 10576   2c2 10626   [,]cicc 11585   ...cfz 11726   ^cexp 12210   sum_csu 13657   EEcee 24608  Cgrccgr 24610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-ee 24611  df-cgr 24613
This theorem is referenced by:  ax5seg  24658
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