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Theorem ax5seglem6 24976
Description: Lemma for ax5seg 24980. 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 1128 . . . 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 9648 . . . . . 6  |-  0  e.  RR
3 1re 9647 . . . . . 6  |-  1  e.  RR
42, 3elicc2i 11707 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
54simp1bi 1024 . . . 4  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
6 resqcl 12349 . . . . 5  |-  ( T  e.  RR  ->  ( T ^ 2 )  e.  RR )
76recnd 9674 . . . 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 1129 . . . 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 11707 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1110simp1bi 1024 . . . 4  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
12 resqcl 12349 . . . . 5  |-  ( S  e.  RR  ->  ( S ^ 2 )  e.  RR )
1312recnd 9674 . . . 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 12193 . . . 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 1054 . . . . . . . 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 1030 . . . . . . 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 24944 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( A `  j )  e.  CC )
1917, 18sylan 474 . . . . . 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 1056 . . . . . . . 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 1030 . . . . . . 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 24944 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( C `  j )  e.  CC )
2321, 22sylan 474 . . . . . 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 9991 . . . . 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 12421 . . . 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 13811 . . 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 1033 . . . 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 1074 . . . 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 1042 . . . 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 1130 . . . 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 24975 . . . 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 1276 . . 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 1037 . . . . . 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 1055 . . . . . . . 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 1057 . . . . . . . 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 1058 . . . . . . . 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 24942 . . . . . . . 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 1270 . . . . . . 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 1030 . . . . . 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 214 . . . . 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 24970 . . . . . 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 1278 . . . . 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 1030 . . . . . 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 1059 . . . . . . 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 1030 . . . . . 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 1131 . . . . . 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 24970 . . . . . 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 1278 . . . . 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 2495 . . . 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 1075 . . . . . 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 1043 . . . . . 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 1044 . . . . . 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 1038 . . . . . 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 24973 . . . . . 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 1297 . . . . 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 6311 . . . 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 2490 . . 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 10255 . 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 1025 . . . . 5  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
605, 59jca 535 . . . 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 1025 . . . . 5  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
6311, 62jca 535 . . . 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 12354 . . 3  |-  ( ( ( T  e.  RR  /\  0  <_  T )  /\  ( S  e.  RR  /\  0  <_  S )
)  ->  ( ( T ^ 2 )  =  ( S ^ 2 )  <->  T  =  S
) )
6661, 64, 65syl2anc 667 . 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 214 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   <.cop 3976   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    x. cmul 9549    <_ cle 9681    - cmin 9865   NNcn 10616   2c2 10666   [,]cicc 11645   ...cfz 11791   ^cexp 12279   sum_csu 13764   EEcee 24930  Cgrccgr 24932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-ee 24933  df-cgr 24935
This theorem is referenced by:  ax5seg  24980
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