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Theorem brcgr 24817
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 4677 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4677 . . 3  |-  <. C ,  D >.  e.  _V
3 eleq1 2492 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
43anbi1d 709 . . . . 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 5872 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
65fveq1d 5874 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 1st `  x ) `  i
)  =  ( ( 1st `  <. A ,  B >. ) `  i
) )
7 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
87fveq1d 5874 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( 2nd `  x ) `  i
)  =  ( ( 2nd `  <. A ,  B >. ) `  i
) )
96, 8oveq12d 6314 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
)  =  ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) )
109oveq1d 6311 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 ) )
1110sumeq2sdv 13737 . . . . . 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 2422 . . . . 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 715 . . . 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 2937 . . 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 2492 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
1615anbi2d 708 . . . . 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 5872 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 1st `  y
)  =  ( 1st `  <. C ,  D >. ) )
1817fveq1d 5874 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 1st `  y ) `  i
)  =  ( ( 1st `  <. C ,  D >. ) `  i
) )
19 fveq2 5872 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( 2nd `  y
)  =  ( 2nd `  <. C ,  D >. ) )
2019fveq1d 5874 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( 2nd `  y ) `  i
)  =  ( ( 2nd `  <. C ,  D >. ) `  i
) )
2118, 20oveq12d 6314 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
)  =  ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) )
2221oveq1d 6311 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) )
2322sumeq2sdv 13737 . . . . . 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 2434 . . . . 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 715 . . . 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 2937 . . 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 24810 . . 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 4735 . 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 4879 . . . . . . . . . . 11  |-  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  ->  D  e.  ( EE `  n ) )
3029ad2antll 733 . . . . . . . . . 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 769 . . . . . . . . . 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 24815 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  n )  /\  D  e.  ( EE `  N ) )  ->  n  =  N )
3330, 31, 32syl2anc 665 . . . . . . . . 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 719 . . . . . . . 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 6304 . . . . . . . . . 10  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
3635sumeq1d 13734 . . . . . . . . 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 13734 . . . . . . . . 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 2442 . . . . . . . 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 6810 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 1st `  <. A ,  B >. )  =  A )
4140fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 1st `  <. A ,  B >. ) `  i )  =  ( A `  i ) )
42 op2ndg 6811 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( 2nd `  <. A ,  B >. )  =  B )
4342fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. A ,  B >. ) `  i )  =  ( B `  i ) )
4441, 43oveq12d 6314 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) )  =  ( ( A `  i )  -  ( B `  i ) ) )
4544oveq1d 6311 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )
4645sumeq2sdv 13737 . . . . . . . . 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 6810 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
4847fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 1st `  <. C ,  D >. ) `  i )  =  ( C `  i ) )
49 op2ndg 6811 . . . . . . . . . . . . 13  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
5049fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( 2nd `  <. C ,  D >. ) `  i )  =  ( D `  i ) )
5148, 50oveq12d 6314 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) )  =  ( ( C `  i )  -  ( D `  i ) ) )
5251oveq1d 6311 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  =  ( ( ( C `  i
)  -  ( D `
 i ) ) ^ 2 ) )
5352sumeq2sdv 13737 . . . . . . . . 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 2443 . . . . . . . 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 730 . . . . . . 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 256 . . . . . 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 210 . . . . 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 606 . . . 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 2915 . . 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 24813 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
6160ad2antll 733 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  ->  N  e.  NN )
62 opelxpi 4877 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
63 opelxpi 4877 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6462, 63anim12i 568 . . . . . . . 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 466 . . . . . . 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 487 . . . . . . 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 534 . . . . . 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 5872 . . . . . . . . . . 11  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
6968sqxpeqd 4871 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
7069eleq2d 2490 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7169eleq2d 2490 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7270, 71anbi12d 715 . . . . . . . 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 715 . . . . . . 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 3179 . . . . . 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 476 . . . . 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 608 . . . 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 193 . 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 260 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 187    /\ wa 370    = wceq 1437    e. wcel 1867   E.wrex 2774   <.cop 3999   class class class wbr 4417    X. cxp 4843   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   1c1 9529    - cmin 9849   NNcn 10598   2c2 10648   ...cfz 11771   ^cexp 12258   sum_csu 13719   EEcee 24805  Cgrccgr 24807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-seq 12200  df-sum 13720  df-ee 24808  df-cgr 24810
This theorem is referenced by:  axcgrrflx  24831  axcgrtr  24832  axcgrid  24833  axsegcon  24844  ax5seglem3  24848  ax5seglem6  24851  ax5seg  24855  axlowdimlem17  24875  ecgrtg  24900
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