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Theorem brcgr 24986
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 4681 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4681 . . 3  |-  <. C ,  D >.  e.  _V
3 eleq1 2528 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
43anbi1d 716 . . . . 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 5892 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
65fveq1d 5894 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 1st `  x ) `  i
)  =  ( ( 1st `  <. A ,  B >. ) `  i
) )
7 fveq2 5892 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
87fveq1d 5894 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 2nd `  x ) `  i
)  =  ( ( 2nd `  <. A ,  B >. ) `  i
) )
96, 8oveq12d 6338 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
)  =  ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) )
109oveq1d 6335 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 ) )
1110sumeq2sdv 13825 . . . . . 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 2464 . . . . 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 722 . . . 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 2913 . . 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 2528 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
1615anbi2d 715 . . . . 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 5892 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 1st `  y
)  =  ( 1st `  <. C ,  D >. ) )
1817fveq1d 5894 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 1st `  y ) `  i
)  =  ( ( 1st `  <. C ,  D >. ) `  i
) )
19 fveq2 5892 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 2nd `  y
)  =  ( 2nd `  <. C ,  D >. ) )
2019fveq1d 5894 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 2nd `  y ) `  i
)  =  ( ( 2nd `  <. C ,  D >. ) `  i
) )
2118, 20oveq12d 6338 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
)  =  ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) )
2221oveq1d 6335 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) )
2322sumeq2sdv 13825 . . . . . 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 2472 . . . . 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 722 . . . 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 2913 . . 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 24979 . . 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 4741 . 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 4890 . . . . . . . . . . 11  |-  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  ->  D  e.  ( EE `  n ) )
3029ad2antll 740 . . . . . . . . . 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 776 . . . . . . . . . 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 24984 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  n )  /\  D  e.  ( EE `  N ) )  ->  n  =  N )
3330, 31, 32syl2anc 671 . . . . . . . . 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 726 . . . . . . . 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 6328 . . . . . . . . . 10  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
3635sumeq1d 13822 . . . . . . . . 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 13822 . . . . . . . . 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 2477 . . . . . . . 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 17 . . . . . . 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 6837 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1st `  <. A ,  B >. )  =  A )
4140fveq1d 5894 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 1st `  <. A ,  B >. ) `  i )  =  ( A `  i ) )
42 op2ndg 6838 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 2nd `  <. A ,  B >. )  =  B )
4342fveq1d 5894 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. A ,  B >. ) `  i )  =  ( B `  i ) )
4441, 43oveq12d 6338 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) )  =  ( ( A `  i )  -  ( B `  i ) ) )
4544oveq1d 6335 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )
4645sumeq2sdv 13825 . . . . . . . . 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 6837 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
4847fveq1d 5894 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 1st `  <. C ,  D >. ) `  i )  =  ( C `  i ) )
49 op2ndg 6838 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
5049fveq1d 5894 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. C ,  D >. ) `  i )  =  ( D `  i ) )
5148, 50oveq12d 6338 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) )  =  ( ( C `  i )  -  ( D `  i ) ) )
5251oveq1d 6335 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  =  ( ( ( C `  i
)  -  ( D `
 i ) ) ^ 2 ) )
5352sumeq2sdv 13825 . . . . . . . . 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 2478 . . . . . . . 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 737 . . . . . . 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 261 . . . . . 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 212 . . . . 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 612 . . . 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 2891 . . 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 24982 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
6160ad2antll 740 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  ->  N  e.  NN )
62 opelxpi 4888 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
63 opelxpi 4888 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6462, 63anim12i 574 . . . . . . . 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 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 ) )  ->  ( <. A ,  B >.  e.  ( ( EE `  N
)  X.  ( EE
`  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) ) ) )
6654biimpar 492 . . . . . . 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 539 . . . . . 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 5892 . . . . . . . . . . 11  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
6968sqxpeqd 4882 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
7069eleq2d 2525 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7169eleq2d 2525 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7270, 71anbi12d 722 . . . . . . . 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 722 . . . . . . 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 3162 . . . . . 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 481 . . . . 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 614 . . . 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 37 . . 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 195 . 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 265 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750   <.cop 3986   class class class wbr 4418    X. cxp 4854   ` cfv 5605  (class class class)co 6320   1stc1st 6823   2ndc2nd 6824   1c1 9571    - cmin 9891   NNcn 10642   2c2 10692   ...cfz 11819   ^cexp 12310   sum_csu 13807   EEcee 24974  Cgrccgr 24976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-seq 12252  df-sum 13808  df-ee 24977  df-cgr 24979
This theorem is referenced by:  axcgrrflx  25000  axcgrtr  25001  axcgrid  25002  axsegcon  25013  ax5seglem3  25017  ax5seglem6  25020  ax5seg  25024  axlowdimlem17  25044  ecgrtg  25069
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