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Theorem brcgr 24600
Description: The binary relationship form of the congruence predicate. The statement  <. A ,  B >.Cgr <. C ,  D >. should be read informally as "the  N dimensional point  A is as far from  B as  C is from  D, or "the line segment  A B is congruent to the line segment  C D. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
brcgr  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
Distinct variable groups:    i, N    A, i    B, i    C, i    D, i

Proof of Theorem brcgr
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4253 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4253 . . 3  |-  <. C ,  D >.  e.  _V
3 eleq1 2356 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
43anbi1d 685 . . . . 5  |-  ( x  =  <. A ,  B >.  ->  ( ( x  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  y  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  <->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n )  X.  ( EE `  n
) ) ) ) )
5 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
65fveq1d 5543 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 1st `  x ) `  i
)  =  ( ( 1st `  <. A ,  B >. ) `  i
) )
7 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
87fveq1d 5543 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 2nd `  x ) `  i
)  =  ( ( 2nd `  <. A ,  B >. ) `  i
) )
96, 8oveq12d 5892 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
)  =  ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) )
109oveq1d 5889 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 ) )
1110sumeq2sdv 12193 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 ) )
1211eqeq1d 2304 . . . . 5  |-  ( x  =  <. A ,  B >.  ->  ( sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 ) ) )
134, 12anbi12d 691 . . . 4  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  x ) `  i
)  -  ( ( 2nd `  x ) `
 i ) ) ^ 2 )  = 
sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
) ^ 2 ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `  i
)  -  ( ( 2nd `  y ) `
 i ) ) ^ 2 ) ) ) )
1413rexbidv 2577 . . 3  |-  ( x  =  <. A ,  B >.  ->  ( E. n  e.  NN  ( ( x  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  y  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `  i
)  -  ( ( 2nd `  y ) `
 i ) ) ^ 2 ) ) ) )
15 eleq1 2356 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
1615anbi2d 684 . . . . 5  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  <->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) ) )
17 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 1st `  y
)  =  ( 1st `  <. C ,  D >. ) )
1817fveq1d 5543 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 1st `  y ) `  i
)  =  ( ( 1st `  <. C ,  D >. ) `  i
) )
19 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 2nd `  y
)  =  ( 2nd `  <. C ,  D >. ) )
2019fveq1d 5543 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 2nd `  y ) `  i
)  =  ( ( 2nd `  <. C ,  D >. ) `  i
) )
2118, 20oveq12d 5892 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
)  =  ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) )
2221oveq1d 5889 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) )
2322sumeq2sdv 12193 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) )
2423eqeq2d 2307 . . . . 5  |-  ( y  =  <. C ,  D >.  ->  ( sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `  i
)  -  ( ( 2nd `  y ) `
 i ) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) )
2516, 24anbi12d 691 . . . 4  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `  i
)  -  ( ( 2nd `  y ) `
 i ) ) ^ 2 ) )  <-> 
( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) ) )
2625rexbidv 2577 . . 3  |-  ( y  =  <. C ,  D >.  ->  ( E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y ) `  i
)  -  ( ( 2nd `  y ) `
 i ) ) ^ 2 ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) ) )
27 df-cgr 24593 . . 3  |- Cgr  =  { <. x ,  y >.  |  E. n  e.  NN  ( ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 ) ) }
281, 2, 14, 26, 27brab 4303 . 2  |-  ( <. A ,  B >.Cgr <. C ,  D >.  <->  E. n  e.  NN  (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) )
29 opelxp2 4739 . . . . . . . . . . 11  |-  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  ->  D  e.  ( EE `  n ) )
3029ad2antll 709 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  D  e.  ( EE `  n
) )
31 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  D  e.  ( EE `  N
) )
32 eedimeq 24598 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  n )  /\  D  e.  ( EE `  N ) )  ->  n  =  N )
3330, 31, 32syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  n  =  N )
3433adantlr 695 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  n  =  N )
35 oveq2 5882 . . . . . . . . . 10  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
3635sumeq1d 12190 . . . . . . . . 9  |-  ( n  =  N  ->  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 ) )
3735sumeq1d 12190 . . . . . . . . 9  |-  ( n  =  N  ->  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) )
3836, 37eqeq12d 2310 . . . . . . . 8  |-  ( n  =  N  ->  ( sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N
) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) ) )
3934, 38syl 15 . . . . . . 7  |-  ( ( ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  ( sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N
) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) ) )
40 op1stg 6148 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1st `  <. A ,  B >. )  =  A )
4140fveq1d 5543 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 1st `  <. A ,  B >. ) `  i )  =  ( A `  i ) )
42 op2ndg 6149 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 2nd `  <. A ,  B >. )  =  B )
4342fveq1d 5543 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. A ,  B >. ) `  i )  =  ( B `  i ) )
4441, 43oveq12d 5892 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) )  =  ( ( A `  i )  -  ( B `  i ) ) )
4544oveq1d 5889 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )
4645sumeq2sdv 12193 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
47 op1stg 6148 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
4847fveq1d 5543 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 1st `  <. C ,  D >. ) `  i )  =  ( C `  i ) )
49 op2ndg 6149 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
5049fveq1d 5543 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. C ,  D >. ) `  i )  =  ( D `  i ) )
5148, 50oveq12d 5892 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) )  =  ( ( C `  i )  -  ( D `  i ) ) )
5251oveq1d 5889 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  =  ( ( ( C `  i
)  -  ( D `
 i ) ) ^ 2 ) )
5352sumeq2sdv 12193 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )
5446, 53eqeqan12d 2311 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N
) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 )  <->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) ) )
5554ad2antrr 706 . . . . . . 7  |-  ( ( ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
5639, 55bitrd 244 . . . . . 6  |-  ( ( ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  ( sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
5756biimpd 198 . . . . 5  |-  ( ( ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  /\  ( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) ) )  ->  ( sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) ) )
5857expimpd 586 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  n  e.  NN )  ->  ( ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) ) )
5958rexlimdva 2680 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) ) )
60 eleenn 24596 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
6160ad2antll 709 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  ->  N  e.  NN )
62 opelxpi 4737 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
63 opelxpi 4737 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6462, 63anim12i 549 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) )  /\  <. C ,  D >.  e.  ( ( EE `  N
)  X.  ( EE
`  N ) ) ) )
6564adantr 451 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )  ->  ( <. A ,  B >.  e.  ( ( EE `  N
)  X.  ( EE
`  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) ) ) )
6654biimpar 471 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) )
6765, 66jca 518 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )  ->  ( ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) ) )
68 fveq2 5541 . . . . . . . . . . 11  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
6968, 68xpeq12d 4730 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
7069eleq2d 2363 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7169eleq2d 2363 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7270, 71anbi12d 691 . . . . . . . 8  |-  ( n  =  N  ->  (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )  <->  ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) ) )
7372, 38anbi12d 691 . . . . . . 7  |-  ( n  =  N  ->  (
( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) ) ) )
7473rspcev 2897 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )  /\  sum_ i  e.  ( 1 ... N
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) ) )
7567, 74sylan2 460 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) ) )
7675exp32 588 . . . 4  |-  ( N  e.  NN  ->  (
( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) ) ) )
7761, 76mpcom 32 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n
) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) ) ^ 2 ) ) ) )
7859, 77impbid 183 . 2  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 ) )  <->  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) ) )
7928, 78syl5bb 248 1  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  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    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   1c1 8754    - cmin 9053   NNcn 9762   2c2 9811   ...cfz 10798   ^cexp 11120   sum_csu 12174   EEcee 24588  Cgrccgr 24590
This theorem is referenced by:  axcgrrflx  24614  axcgrtr  24615  axcgrid  24616  axsegcon  24627  ax5seglem3  24631  ax5seglem6  24634  ax5seg  24638  axlowdimlem17  24658
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  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
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-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-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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-sum 12175  df-ee 24591  df-cgr 24593
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