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Theorem ecgrtg 24059
Description: The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
ecgrtg.1  |-  ( ph  ->  N  e.  NN )
ecgrtg.2  |-  P  =  ( Base `  (EEG `  N ) )
ecgrtg.3  |-  .-  =  ( dist `  (EEG `  N
) )
ecgrtg.a  |-  ( ph  ->  A  e.  P )
ecgrtg.b  |-  ( ph  ->  B  e.  P )
ecgrtg.c  |-  ( ph  ->  C  e.  P )
ecgrtg.d  |-  ( ph  ->  D  e.  P )
Assertion
Ref Expression
ecgrtg  |-  ( ph  ->  ( <. A ,  B >.Cgr
<. C ,  D >.  <->  ( A  .-  B )  =  ( C  .-  D
) ) )

Proof of Theorem ecgrtg
Dummy variables  x  i  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecgrtg.a . . . 4  |-  ( ph  ->  A  e.  P )
2 ecgrtg.1 . . . . . 6  |-  ( ph  ->  N  e.  NN )
3 eengbas 24057 . . . . . 6  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( Base `  (EEG `  N ) ) )
42, 3syl 16 . . . . 5  |-  ( ph  ->  ( EE `  N
)  =  ( Base `  (EEG `  N )
) )
5 ecgrtg.2 . . . . 5  |-  P  =  ( Base `  (EEG `  N ) )
64, 5syl6eqr 2526 . . . 4  |-  ( ph  ->  ( EE `  N
)  =  P )
71, 6eleqtrrd 2558 . . 3  |-  ( ph  ->  A  e.  ( EE
`  N ) )
8 ecgrtg.b . . . 4  |-  ( ph  ->  B  e.  P )
98, 6eleqtrrd 2558 . . 3  |-  ( ph  ->  B  e.  ( EE
`  N ) )
10 ecgrtg.c . . . 4  |-  ( ph  ->  C  e.  P )
1110, 6eleqtrrd 2558 . . 3  |-  ( ph  ->  C  e.  ( EE
`  N ) )
12 ecgrtg.d . . . 4  |-  ( ph  ->  D  e.  P )
1312, 6eleqtrrd 2558 . . 3  |-  ( ph  ->  D  e.  ( EE
`  N ) )
14 brcgr 23976 . . 3  |-  ( ( ( 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 ) ) )
157, 9, 11, 13, 14syl22anc 1229 . 2  |-  ( ph  ->  ( <. 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 ) ) )
16 dsid 14662 . . . . . . 7  |-  dist  = Slot  ( dist `  ndx )
17 fvex 5876 . . . . . . . 8  |-  (EEG `  N )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ph  ->  (EEG `  N )  e.  _V )
19 eengstr 24056 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (EEG `  N ) Struct  <. 1 , ; 1 7 >. )
202, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  (EEG `  N ) Struct  <.
1 , ; 1 7 >. )
21 isstruct 14503 . . . . . . . . . 10  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  <->  ( (
1  e.  NN  /\ ; 1 7  e.  NN  /\  1  <_ ; 1
7 )  /\  Fun  ( (EEG `  N )  \  { (/) } )  /\  dom  (EEG `  N )  C_  ( 1 ...; 1 7 ) ) )
2221simp2bi 1012 . . . . . . . . 9  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  Fun  ( (EEG `  N )  \  { (/) } ) )
2320, 22syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  ( (EEG `  N )  \  { (/)
} ) )
24 structcnvcnv 14504 . . . . . . . . . 10  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2520, 24syl 16 . . . . . . . . 9  |-  ( ph  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2625funeqd 5609 . . . . . . . 8  |-  ( ph  ->  ( Fun  `' `' (EEG `  N )  <->  Fun  ( (EEG
`  N )  \  { (/) } ) ) )
2723, 26mpbird 232 . . . . . . 7  |-  ( ph  ->  Fun  `' `' (EEG
`  N ) )
28 opex 4711 . . . . . . . . . 10  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  _V
2928prid2 4136 . . . . . . . . 9  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }
30 elun1 3671 . . . . . . . . 9  |-  ( <.
( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  ->  <.
( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
3129, 30ax-mp 5 . . . . . . . 8  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } )
32 eengv 24055 . . . . . . . . 9  |-  ( N  e.  NN  ->  (EEG `  N )  =  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
332, 32syl 16 . . . . . . . 8  |-  ( ph  ->  (EEG `  N )  =  ( { <. (
Base `  ndx ) ,  ( EE `  N
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
3431, 33syl5eleqr 2562 . . . . . . 7  |-  ( ph  -> 
<. ( dist `  ndx ) ,  ( x  e.  ( EE `  N
) ,  y  e.  ( EE `  N
)  |->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  (EEG
`  N ) )
35 fvex 5876 . . . . . . . . 9  |-  ( EE
`  N )  e. 
_V
3635, 35mpt2ex 6861 . . . . . . . 8  |-  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) )  e. 
_V
3736a1i 11 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) )  e.  _V )
3816, 18, 27, 34, 37strfv2d 14525 . . . . . 6  |-  ( ph  ->  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) )  =  (
dist `  (EEG `  N
) ) )
39 ecgrtg.3 . . . . . 6  |-  .-  =  ( dist `  (EEG `  N
) )
4038, 39syl6reqr 2527 . . . . 5  |-  ( ph  ->  .-  =  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) ) )
41 simplrl 759 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  x  =  A )
4241fveq1d 5868 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
x `  i )  =  ( A `  i ) )
43 simplrr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  y  =  B )
4443fveq1d 5868 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( B `  i ) )
4542, 44oveq12d 6303 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( A `
 i )  -  ( B `  i ) ) )
4645oveq1d 6300 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
4746sumeq2dv 13491 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
48 sumex 13476 . . . . . 6  |-  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  e.  _V
4948a1i 11 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  e.  _V )
5040, 47, 7, 9, 49ovmpt2d 6415 . . . 4  |-  ( ph  ->  ( A  .-  B
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
51 eqcom 2476 . . . . 5  |-  ( ( A  .-  B )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  ( A  .-  B ) )
5251imbi2i 312 . . . 4  |-  ( (
ph  ->  ( A  .-  B )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 ) )  <->  ( ph  ->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  ( A  .-  B ) ) )
5350, 52mpbi 208 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  ( A  .-  B ) )
54 simplrl 759 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  x  =  C )
5554fveq1d 5868 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
x `  i )  =  ( C `  i ) )
56 simplrr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  y  =  D )
5756fveq1d 5868 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( D `  i ) )
5855, 57oveq12d 6303 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( C `
 i )  -  ( D `  i ) ) )
5958oveq1d 6300 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
6059sumeq2dv 13491 . . . . 5  |-  ( (
ph  /\  ( x  =  C  /\  y  =  D ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )
61 sumex 13476 . . . . . 6  |-  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  e.  _V
6261a1i 11 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  e.  _V )
6340, 60, 11, 13, 62ovmpt2d 6415 . . . 4  |-  ( ph  ->  ( C  .-  D
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
64 eqcom 2476 . . . . 5  |-  ( ( C  .-  D )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  <->  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  =  ( C  .-  D ) )
6564imbi2i 312 . . . 4  |-  ( (
ph  ->  ( C  .-  D )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 ) )  <->  ( ph  ->  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  =  ( C  .-  D ) ) )
6663, 65mpbi 208 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  =  ( C  .-  D ) )
6753, 66eqeq12d 2489 . 2  |-  ( ph  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  <-> 
( A  .-  B
)  =  ( C 
.-  D ) ) )
6815, 67bitrd 253 1  |-  ( ph  ->  ( <. A ,  B >.Cgr
<. C ,  D >.  <->  ( A  .-  B )  =  ( C  .-  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   <.cop 4033   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   Fun wfun 5582   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1c1 9494    <_ cle 9630    - cmin 9806   NNcn 10537   2c2 10586   7c7 10591  ;cdc 10977   ...cfz 11673   ^cexp 12135   sum_csu 13474   Struct cstr 14489   ndxcnx 14490   Basecbs 14493   distcds 14567  Itvcitv 23657  LineGclng 23658   EEcee 23964    Btwn cbtwn 23965  Cgrccgr 23966  EEGceeng 24053
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-seq 12077  df-sum 13475  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-ds 14580  df-itv 23659  df-lng 23660  df-ee 23967  df-cgr 23969  df-eeng 24054
This theorem is referenced by:  eengtrkg  24061
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