Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axsegconlem9 Unicode version

Theorem axsegconlem9 24625
Description: Lemma for axsegcon 24627. Show that  B F is congruent to  C D. (Contributed by Scott Fenton, 19-Sep-2013.)
Hypotheses
Ref Expression
axsegconlem2.1  |-  S  = 
sum_ p  e.  (
1 ... N ) ( ( ( A `  p )  -  ( B `  p )
) ^ 2 )
axsegconlem7.2  |-  T  = 
sum_ p  e.  (
1 ... N ) ( ( ( C `  p )  -  ( D `  p )
) ^ 2 )
axsegconlem8.3  |-  F  =  ( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )
Assertion
Ref Expression
axsegconlem9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( B `  i )  -  ( F `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) )
Distinct variable groups:    A, p    B, p    C, p    D, p    N, p    A, i, k    B, i, k    C, i, k    D, i, k    i, N, k    S, i, k    T, i, k    i, p
Allowed substitution hints:    S( p)    T( p)    F( i, k, p)

Proof of Theorem axsegconlem9
StepHypRef Expression
1 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  i  ->  ( B `  k )  =  ( B `  i ) )
21oveq2d 5890 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  =  ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) ) )
3 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  i  ->  ( A `  k )  =  ( A `  i ) )
43oveq2d 5890 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( sqr `  T
)  x.  ( A `
 k ) )  =  ( ( sqr `  T )  x.  ( A `  i )
) )
52, 4oveq12d 5892 . . . . . . . . . 10  |-  ( k  =  i  ->  (
( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 k ) )  -  ( ( sqr `  T )  x.  ( A `  k )
) )  =  ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 i ) )  -  ( ( sqr `  T )  x.  ( A `  i )
) ) )
65oveq1d 5889 . . . . . . . . 9  |-  ( k  =  i  ->  (
( ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  k )
)  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S
) )  =  ( ( ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  i )
)  -  ( ( sqr `  T )  x.  ( A `  i ) ) )  /  ( sqr `  S
) ) )
7 axsegconlem8.3 . . . . . . . . 9  |-  F  =  ( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )
8 ovex 5899 . . . . . . . . 9  |-  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 i ) )  -  ( ( sqr `  T )  x.  ( A `  i )
) )  /  ( sqr `  S ) )  e.  _V
96, 7, 8fvmpt 5618 . . . . . . . 8  |-  ( i  e.  ( 1 ... N )  ->  ( F `  i )  =  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) )  /  ( sqr `  S ) ) )
109adantl 452 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( F `  i )  =  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) )  /  ( sqr `  S ) ) )
1110oveq2d 5890 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( B `  i
)  -  ( F `
 i ) )  =  ( ( B `
 i )  -  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) )  /  ( sqr `  S ) ) ) )
12 axsegconlem2.1 . . . . . . . . . . . . 13  |-  S  = 
sum_ p  e.  (
1 ... N ) ( ( ( A `  p )  -  ( B `  p )
) ^ 2 )
1312axsegconlem4 24620 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( sqr `  S
)  e.  RR )
14133adant3 975 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  ( sqr `  S )  e.  RR )
1514ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( sqr `  S )  e.  RR )
16 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
17 fveere 24601 . . . . . . . . . . 11  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
1816, 17sylan 457 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  RR )
1915, 18remulcld 8879 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  x.  ( B `
 i ) )  e.  RR )
2019recnd 8877 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  x.  ( B `
 i ) )  e.  CC )
21 axsegconlem7.2 . . . . . . . . . . . . . 14  |-  T  = 
sum_ p  e.  (
1 ... N ) ( ( ( C `  p )  -  ( D `  p )
) ^ 2 )
2221axsegconlem4 24620 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( sqr `  T
)  e.  RR )
23 readdcl 8836 . . . . . . . . . . . . 13  |-  ( ( ( sqr `  S
)  e.  RR  /\  ( sqr `  T )  e.  RR )  -> 
( ( sqr `  S
)  +  ( sqr `  T ) )  e.  RR )
2414, 22, 23syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( sqr `  S
)  +  ( sqr `  T ) )  e.  RR )
2524adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  +  ( sqr `  T ) )  e.  RR )
2625, 18remulcld 8879 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  e.  RR )
2722ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( sqr `  T )  e.  RR )
28 simpl1 958 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
29 fveere 24601 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
3028, 29sylan 457 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  RR )
3127, 30remulcld 8879 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( A `
 i ) )  e.  RR )
3226, 31resubcld 9227 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 i ) )  -  ( ( sqr `  T )  x.  ( A `  i )
) )  e.  RR )
3332recnd 8877 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 i ) )  -  ( ( sqr `  T )  x.  ( A `  i )
) )  e.  CC )
3415recnd 8877 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( sqr `  S )  e.  CC )
3512axsegconlem6 24622 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  0  <  ( sqr `  S
) )
3635gt0ne0d 9353 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  ( sqr `  S )  =/=  0 )
3736ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( sqr `  S )  =/=  0 )
3820, 33, 34, 37divsubdird 9591 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  x.  ( B `  i )
)  -  ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) ) )  /  ( sqr `  S ) )  =  ( ( ( ( sqr `  S
)  x.  ( B `
 i ) )  /  ( sqr `  S
) )  -  (
( ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  i )
)  -  ( ( sqr `  T )  x.  ( A `  i ) ) )  /  ( sqr `  S
) ) ) )
3926recnd 8877 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  e.  CC )
4031recnd 8877 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( A `
 i ) )  e.  CC )
4120, 39, 40subsubd 9201 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  x.  ( B `
 i ) )  -  ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) ) )  =  ( ( ( ( sqr `  S )  x.  ( B `  i )
)  -  ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) ) )  +  ( ( sqr `  T
)  x.  ( A `
 i ) ) ) )
4227recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( sqr `  T )  e.  CC )
4318renegcld 9226 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  -u ( B `  i )  e.  RR )
4443recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  -u ( B `  i )  e.  CC )
4530recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( A `  i )  e.  CC )
4642, 44, 45adddid 8875 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( -u ( B `  i )  +  ( A `  i ) ) )  =  ( ( ( sqr `  T )  x.  -u ( B `  i ) )  +  ( ( sqr `  T
)  x.  ( A `
 i ) ) ) )
4744, 45addcomd 9030 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( -u ( B `  i
)  +  ( A `
 i ) )  =  ( ( A `
 i )  + 
-u ( B `  i ) ) )
4818recnd 8877 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( B `  i )  e.  CC )
4945, 48negsubd 9179 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  +  -u ( B `  i )
)  =  ( ( A `  i )  -  ( B `  i ) ) )
5047, 49eqtrd 2328 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( -u ( B `  i
)  +  ( A `
 i ) )  =  ( ( A `
 i )  -  ( B `  i ) ) )
5150oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( -u ( B `  i )  +  ( A `  i ) ) )  =  ( ( sqr `  T )  x.  (
( A `  i
)  -  ( B `
 i ) ) ) )
5225recnd 8877 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  +  ( sqr `  T ) )  e.  CC )
5352, 34negsubdi2d 9189 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  -u (
( ( sqr `  S
)  +  ( sqr `  T ) )  -  ( sqr `  S ) )  =  ( ( sqr `  S )  -  ( ( sqr `  S )  +  ( sqr `  T ) ) ) )
5434, 42pncan2d 9175 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  +  ( sqr `  T ) )  -  ( sqr `  S ) )  =  ( sqr `  T ) )
5554negeqd 9062 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  -u (
( ( sqr `  S
)  +  ( sqr `  T ) )  -  ( sqr `  S ) )  =  -u ( sqr `  T ) )
5653, 55eqtr3d 2330 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  -  ( ( sqr `  S )  +  ( sqr `  T
) ) )  = 
-u ( sqr `  T
) )
5756oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  -  ( ( sqr `  S )  +  ( sqr `  T
) ) )  x.  ( B `  i
) )  =  (
-u ( sqr `  T
)  x.  ( B `
 i ) ) )
5834, 52, 48subdird 9252 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  -  ( ( sqr `  S )  +  ( sqr `  T
) ) )  x.  ( B `  i
) )  =  ( ( ( sqr `  S
)  x.  ( B `
 i ) )  -  ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  i )
) ) )
59 mulneg12 9234 . . . . . . . . . . . . 13  |-  ( ( ( sqr `  T
)  e.  CC  /\  ( B `  i )  e.  CC )  -> 
( -u ( sqr `  T
)  x.  ( B `
 i ) )  =  ( ( sqr `  T )  x.  -u ( B `  i )
) )
6042, 48, 59syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( -u ( sqr `  T
)  x.  ( B `
 i ) )  =  ( ( sqr `  T )  x.  -u ( B `  i )
) )
6157, 58, 603eqtr3rd 2337 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  -u ( B `  i )
)  =  ( ( ( sqr `  S
)  x.  ( B `
 i ) )  -  ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  i )
) ) )
6261oveq1d 5889 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  T
)  x.  -u ( B `  i )
)  +  ( ( sqr `  T )  x.  ( A `  i ) ) )  =  ( ( ( ( sqr `  S
)  x.  ( B `
 i ) )  -  ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  i )
) )  +  ( ( sqr `  T
)  x.  ( A `
 i ) ) ) )
6346, 51, 623eqtr3rd 2337 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  x.  ( B `  i )
)  -  ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) ) )  +  ( ( sqr `  T
)  x.  ( A `
 i ) ) )  =  ( ( sqr `  T )  x.  ( ( A `
 i )  -  ( B `  i ) ) ) )
6441, 63eqtrd 2328 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  x.  ( B `
 i ) )  -  ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) ) )  =  ( ( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) ) )
6564oveq1d 5889 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  x.  ( B `  i )
)  -  ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) ) )  /  ( sqr `  S ) )  =  ( ( ( sqr `  T )  x.  ( ( A `
 i )  -  ( B `  i ) ) )  /  ( sqr `  S ) ) )
6648, 34, 37divcan3d 9557 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  x.  ( B `
 i ) )  /  ( sqr `  S
) )  =  ( B `  i ) )
6766oveq1d 5889 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  x.  ( B `  i )
)  /  ( sqr `  S ) )  -  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) )  /  ( sqr `  S ) ) )  =  ( ( B `
 i )  -  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  i
) )  -  (
( sqr `  T
)  x.  ( A `
 i ) ) )  /  ( sqr `  S ) ) ) )
6838, 65, 673eqtr3rd 2337 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( B `  i
)  -  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 i ) )  -  ( ( sqr `  T )  x.  ( A `  i )
) )  /  ( sqr `  S ) ) )  =  ( ( ( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) )  /  ( sqr `  S
) ) )
6911, 68eqtrd 2328 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( B `  i
)  -  ( F `
 i ) )  =  ( ( ( sqr `  T )  x.  ( ( A `
 i )  -  ( B `  i ) ) )  /  ( sqr `  S ) ) )
7069oveq1d 5889 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( B `  i )  -  ( F `  i )
) ^ 2 )  =  ( ( ( ( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) )  /  ( sqr `  S
) ) ^ 2 ) )
7130, 18resubcld 9227 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
7227, 71remulcld 8879 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) )  e.  RR )
7372recnd 8877 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) )  e.  CC )
7473, 34, 37sqdivd 11274 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  T )  x.  (
( A `  i
)  -  ( B `
 i ) ) )  /  ( sqr `  S ) ) ^
2 )  =  ( ( ( ( sqr `  T )  x.  (
( A `  i
)  -  ( B `
 i ) ) ) ^ 2 )  /  ( ( sqr `  S ) ^ 2 ) ) )
7571recnd 8877 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  CC )
7642, 75sqmuld 11273 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) ) ^ 2 )  =  ( ( ( sqr `  T ) ^ 2 )  x.  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) ) )
7721axsegconlem2 24618 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  T  e.  RR )
7877ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  T  e.  RR )
7921axsegconlem3 24619 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
0  <_  T )
8079ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  0  <_  T )
81 resqrth 11757 . . . . . . . 8  |-  ( ( T  e.  RR  /\  0  <_  T )  -> 
( ( sqr `  T
) ^ 2 )  =  T )
8278, 80, 81syl2anc 642 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  T
) ^ 2 )  =  T )
8382oveq1d 5889 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  T
) ^ 2 )  x.  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  =  ( T  x.  (
( ( A `  i )  -  ( B `  i )
) ^ 2 ) ) )
8476, 83eqtrd 2328 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( sqr `  T
)  x.  ( ( A `  i )  -  ( B `  i ) ) ) ^ 2 )  =  ( T  x.  (
( ( A `  i )  -  ( B `  i )
) ^ 2 ) ) )
8512axsegconlem2 24618 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  S  e.  RR )
8612axsegconlem3 24619 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
0  <_  S )
87 resqrth 11757 . . . . . . . 8  |-  ( ( S  e.  RR  /\  0  <_  S )  -> 
( ( sqr `  S
) ^ 2 )  =  S )
8885, 86, 87syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( sqr `  S
) ^ 2 )  =  S )
89883adant3 975 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  (
( sqr `  S
) ^ 2 )  =  S )
9089ad2antrr 706 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( sqr `  S
) ^ 2 )  =  S )
9184, 90oveq12d 5892 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  T )  x.  (
( A `  i
)  -  ( B `
 i ) ) ) ^ 2 )  /  ( ( sqr `  S ) ^ 2 ) )  =  ( ( T  x.  (
( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  /  S ) )
9270, 74, 913eqtrd 2332 . . 3  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( B `  i )  -  ( F `  i )
) ^ 2 )  =  ( ( T  x.  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  /  S ) )
9392sumeq2dv 12192 . 2  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( B `  i )  -  ( F `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( T  x.  (
( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  /  S ) )
94 fzfid 11051 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
1 ... N )  e. 
Fin )
9577adantl 452 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  T  e.  RR )
9695recnd 8877 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  T  e.  CC )
9771resqcld 11287 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
9897recnd 8877 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  CC )
9994, 96, 98fsummulc2 12262 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( T  x.  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  = 
sum_ i  e.  ( 1 ... N ) ( T  x.  (
( ( A `  i )  -  ( B `  i )
) ^ 2 ) ) )
10099oveq1d 5889 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( T  x.  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  /  S )  =  ( sum_ i  e.  ( 1 ... N
) ( T  x.  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )  /  S ) )
101 fveq2 5541 . . . . . . . . 9  |-  ( p  =  i  ->  ( C `  p )  =  ( C `  i ) )
102 fveq2 5541 . . . . . . . . 9  |-  ( p  =  i  ->  ( D `  p )  =  ( D `  i ) )
103101, 102oveq12d 5892 . . . . . . . 8  |-  ( p  =  i  ->  (
( C `  p
)  -  ( D `
 p ) )  =  ( ( C `
 i )  -  ( D `  i ) ) )
104103oveq1d 5889 . . . . . . 7  |-  ( p  =  i  ->  (
( ( C `  p )  -  ( D `  p )
) ^ 2 )  =  ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
105104cbvsumv 12185 . . . . . 6  |-  sum_ p  e.  ( 1 ... N
) ( ( ( C `  p )  -  ( D `  p ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )
10621, 105eqtri 2316 . . . . 5  |-  T  = 
sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )
107 fveq2 5541 . . . . . . . . 9  |-  ( i  =  p  ->  ( A `  i )  =  ( A `  p ) )
108 fveq2 5541 . . . . . . . . 9  |-  ( i  =  p  ->  ( B `  i )  =  ( B `  p ) )
109107, 108oveq12d 5892 . . . . . . . 8  |-  ( i  =  p  ->  (
( A `  i
)  -  ( B `
 i ) )  =  ( ( A `
 p )  -  ( B `  p ) ) )
110109oveq1d 5889 . . . . . . 7  |-  ( i  =  p  ->  (
( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  ( ( ( A `  p )  -  ( B `  p ) ) ^
2 ) )
111110cbvsumv 12185 . . . . . 6  |-  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ p  e.  ( 1 ... N ) ( ( ( A `  p
)  -  ( B `
 p ) ) ^ 2 )
112111, 12eqtr4i 2319 . . . . 5  |-  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  S
113106, 112oveq12i 5886 . . . 4  |-  ( T  x.  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )  =  ( sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  x.  S )
114 eqid 2296 . . . . . . . . . 10  |-  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )
115114axsegconlem2 24618 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  e.  RR )
1161153adant3 975 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  e.  RR )
117116adantr 451 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  e.  RR )
11895, 117remulcld 8879 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( T  x.  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  e.  RR )
119118recnd 8877 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( T  x.  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  e.  CC )
120 eqid 2296 . . . . . . . 8  |-  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )
121120axsegconlem2 24618 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  e.  RR )
122121adantl 452 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  e.  RR )
123122recnd 8877 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  e.  CC )
124853adant3 975 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  S  e.  RR )
125124adantr 451 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  S  e.  RR )
126125recnd 8877 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  S  e.  CC )
127863adant3 975 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  0  <_  S )
128 sqr00 11765 . . . . . . . . 9  |-  ( ( S  e.  RR  /\  0  <_  S )  -> 
( ( sqr `  S
)  =  0  <->  S  =  0 ) )
129128necon3bid 2494 . . . . . . . 8  |-  ( ( S  e.  RR  /\  0  <_  S )  -> 
( ( sqr `  S
)  =/=  0  <->  S  =/=  0 ) )
130124, 127, 129syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  (
( sqr `  S
)  =/=  0  <->  S  =/=  0 ) )
13136, 130mpbid 201 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  S  =/=  0 )
132131adantr 451 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  S  =/=  0 )
133119, 123, 126, 132divmul3d 9586 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( ( T  x.  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  /  S )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  <-> 
( T  x.  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  =  ( sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  x.  S ) ) )
134113, 133mpbiri 224 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( T  x.  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  /  S )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )
13578, 97remulcld 8879 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( T  x.  ( (
( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )  e.  RR )
136135recnd 8877 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  i  e.  ( 1 ... N
) )  ->  ( T  x.  ( (
( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )  e.  CC )
13794, 126, 136, 132fsumdivc 12264 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( sum_ i  e.  ( 1 ... N ) ( T  x.  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )  /  S )  = 
sum_ i  e.  ( 1 ... N ) ( ( T  x.  ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )  /  S ) )
138100, 134, 1373eqtr3rd 2337 . 2  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( T  x.  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )  /  S )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) )
13993, 138eqtrd 2328 1  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( B `  i )  -  ( F `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   ...cfz 10798   ^cexp 11120   sqrcsqr 11734   sum_csu 12174   EEcee 24588
This theorem is referenced by:  axsegcon  24627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-ee 24591
  Copyright terms: Public domain W3C validator