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Theorem brcgr 25844
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 4430 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4430 . . 3  |-  <. C ,  D >.  e.  _V
3 eleq1 2498 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
43anbi1d 687 . . . . 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 5731 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
65fveq1d 5733 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 1st `  x ) `  i
)  =  ( ( 1st `  <. A ,  B >. ) `  i
) )
7 fveq2 5731 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
87fveq1d 5733 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 2nd `  x ) `  i
)  =  ( ( 2nd `  <. A ,  B >. ) `  i
) )
96, 8oveq12d 6102 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
)  =  ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) )
109oveq1d 6099 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 ) )
1110sumeq2sdv 12503 . . . . . 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 2446 . . . . 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 693 . . . 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 2728 . . 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 2498 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
1615anbi2d 686 . . . . 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 5731 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 1st `  y
)  =  ( 1st `  <. C ,  D >. ) )
1817fveq1d 5733 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 1st `  y ) `  i
)  =  ( ( 1st `  <. C ,  D >. ) `  i
) )
19 fveq2 5731 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 2nd `  y
)  =  ( 2nd `  <. C ,  D >. ) )
2019fveq1d 5733 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 2nd `  y ) `  i
)  =  ( ( 2nd `  <. C ,  D >. ) `  i
) )
2118, 20oveq12d 6102 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
)  =  ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) )
2221oveq1d 6099 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) )
2322sumeq2sdv 12503 . . . . . 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 2449 . . . . 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 693 . . . 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 2728 . . 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 25837 . . 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 4480 . 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 4915 . . . . . . . . . . 11  |-  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  ->  D  e.  ( EE `  n ) )
3029ad2antll 711 . . . . . . . . . 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 739 . . . . . . . . . 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 25842 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  n )  /\  D  e.  ( EE `  N ) )  ->  n  =  N )
3330, 31, 32syl2anc 644 . . . . . . . . 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 697 . . . . . . . 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 6092 . . . . . . . . . 10  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
3635sumeq1d 12500 . . . . . . . . 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 12500 . . . . . . . . 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 2452 . . . . . . . 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 16 . . . . . . 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 6362 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1st `  <. A ,  B >. )  =  A )
4140fveq1d 5733 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 1st `  <. A ,  B >. ) `  i )  =  ( A `  i ) )
42 op2ndg 6363 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 2nd `  <. A ,  B >. )  =  B )
4342fveq1d 5733 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. A ,  B >. ) `  i )  =  ( B `  i ) )
4441, 43oveq12d 6102 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) )  =  ( ( A `  i )  -  ( B `  i ) ) )
4544oveq1d 6099 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )
4645sumeq2sdv 12503 . . . . . . . . 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 6362 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
4847fveq1d 5733 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 1st `  <. C ,  D >. ) `  i )  =  ( C `  i ) )
49 op2ndg 6363 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
5049fveq1d 5733 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. C ,  D >. ) `  i )  =  ( D `  i ) )
5148, 50oveq12d 6102 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) )  =  ( ( C `  i )  -  ( D `  i ) ) )
5251oveq1d 6099 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  =  ( ( ( C `  i
)  -  ( D `
 i ) ) ^ 2 ) )
5352sumeq2sdv 12503 . . . . . . . . 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 2453 . . . . . . . 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 708 . . . . . . 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 246 . . . . . 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 200 . . . . 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 588 . . . 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 2832 . . 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 25840 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
6160ad2antll 711 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  ->  N  e.  NN )
62 opelxpi 4913 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
63 opelxpi 4913 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6462, 63anim12i 551 . . . . . . . 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 453 . . . . . . 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 473 . . . . . . 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 520 . . . . . 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 5731 . . . . . . . . . . 11  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
6968, 68xpeq12d 4906 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
7069eleq2d 2505 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7169eleq2d 2505 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7270, 71anbi12d 693 . . . . . . . 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 693 . . . . . . 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 3054 . . . . . 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 462 . . . . 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 590 . . . 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 35 . . 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 185 . 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 250 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   <.cop 3819   class class class wbr 4215    X. cxp 4879   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   1c1 8996    - cmin 9296   NNcn 10005   2c2 10054   ...cfz 11048   ^cexp 11387   sum_csu 12484   EEcee 25832  Cgrccgr 25834
This theorem is referenced by:  axcgrrflx  25858  axcgrtr  25859  axcgrid  25860  axsegcon  25871  ax5seglem3  25875  ax5seglem6  25878  ax5seg  25882  axlowdimlem17  25902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-seq 11329  df-sum 12485  df-ee 25835  df-cgr 25837
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