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Theorem brcgr 24026
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 4717 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4717 . . 3  |-  <. C ,  D >.  e.  _V
3 eleq1 2539 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
43anbi1d 704 . . . . 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 6313 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( ( 1st `  x ) `
 i )  -  ( ( 2nd `  x
) `  i )
)  =  ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) )
109oveq1d 6310 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( ( ( 1st `  x
) `  i )  -  ( ( 2nd `  x ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. A ,  B >. ) `  i )  -  (
( 2nd `  <. A ,  B >. ) `  i ) ) ^
2 ) )
1110sumeq2sdv 13506 . . . . . 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 2469 . . . . 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 710 . . . 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 2978 . . 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 2539 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
1615anbi2d 703 . . . . 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 6313 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( ( 1st `  y ) `
 i )  -  ( ( 2nd `  y
) `  i )
)  =  ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) )
2221oveq1d 6310 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( ( ( 1st `  y
) `  i )  -  ( ( 2nd `  y ) `  i
) ) ^ 2 )  =  ( ( ( ( 1st `  <. C ,  D >. ) `  i )  -  (
( 2nd `  <. C ,  D >. ) `  i ) ) ^
2 ) )
2322sumeq2sdv 13506 . . . . . 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 2481 . . . . 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 710 . . . 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 2978 . . 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 24019 . . 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 4776 . 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 5039 . . . . . . . . . . 11  |-  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  ->  D  e.  ( EE `  n ) )
3029ad2antll 728 . . . . . . . . . 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 760 . . . . . . . . . 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 24024 . . . . . . . . . 10  |-  ( ( D  e.  ( EE
`  n )  /\  D  e.  ( EE `  N ) )  ->  n  =  N )
3330, 31, 32syl2anc 661 . . . . . . . . 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 714 . . . . . . . 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 6303 . . . . . . . . . 10  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
3635sumeq1d 13503 . . . . . . . . 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 13503 . . . . . . . . 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 2489 . . . . . . . 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 6807 . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . 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 6313 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. A ,  B >. ) `  i )  -  ( ( 2nd `  <. A ,  B >. ) `  i ) )  =  ( ( A `  i )  -  ( B `  i ) ) )
4544oveq1d 6310 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. A ,  B >. ) `  i
)  -  ( ( 2nd `  <. A ,  B >. ) `  i
) ) ^ 2 )  =  ( ( ( A `  i
)  -  ( B `
 i ) ) ^ 2 ) )
4645sumeq2sdv 13506 . . . . . . . . 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 6807 . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . 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 6313 . . . . . . . . . . 11  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( 1st `  <. C ,  D >. ) `  i )  -  ( ( 2nd `  <. C ,  D >. ) `  i ) )  =  ( ( C `  i )  -  ( D `  i ) ) )
5251oveq1d 6310 . . . . . . . . . 10  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( ( ( ( 1st `  <. C ,  D >. ) `  i
)  -  ( ( 2nd `  <. C ,  D >. ) `  i
) ) ^ 2 )  =  ( ( ( C `  i
)  -  ( D `
 i ) ) ^ 2 ) )
5352sumeq2sdv 13506 . . . . . . . . 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 2490 . . . . . . . 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 725 . . . . . . 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 253 . . . . . 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 207 . . . . 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 603 . . . 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 2959 . . 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 24022 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
6160ad2antll 728 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  ->  N  e.  NN )
62 opelxpi 5037 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
63 opelxpi 5037 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6462, 63anim12i 566 . . . . . . . 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 465 . . . . . . 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 485 . . . . . . 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 532 . . . . . 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
) )
6968, 68xpeq12d 5030 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
7069eleq2d 2537 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7169eleq2d 2537 . . . . . . . . 9  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
7270, 71anbi12d 710 . . . . . . . 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 710 . . . . . . 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 3219 . . . . . 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 474 . . . . 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 605 . . . 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 36 . . 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 191 . 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 257 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   <.cop 4039   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   1c1 9505    - cmin 9817   NNcn 10548   2c2 10597   ...cfz 11684   ^cexp 12146   sum_csu 13488   EEcee 24014  Cgrccgr 24016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088  df-sum 13489  df-ee 24017  df-cgr 24019
This theorem is referenced by:  axcgrrflx  24040  axcgrtr  24041  axcgrid  24042  axsegcon  24053  ax5seglem3  24057  ax5seglem6  24060  ax5seg  24064  axlowdimlem17  24084  ecgrtg  24109
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