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Theorem ecgrtg 23229
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 23227 . . . . . 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 2493 . . . 4  |-  ( ph  ->  ( EE `  N
)  =  P )
71, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  A  e.  ( EE
`  N ) )
8 ecgrtg.b . . . 4  |-  ( ph  ->  B  e.  P )
98, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  B  e.  ( EE
`  N ) )
10 ecgrtg.c . . . 4  |-  ( ph  ->  C  e.  P )
1110, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  C  e.  ( EE
`  N ) )
12 ecgrtg.d . . . 4  |-  ( ph  ->  D  e.  P )
1312, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  D  e.  ( EE
`  N ) )
14 brcgr 23146 . . 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 1219 . 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 14342 . . . . . . 7  |-  dist  = Slot  ( dist `  ndx )
17 fvex 5701 . . . . . . . 8  |-  (EEG `  N )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ph  ->  (EEG `  N )  e.  _V )
19 eengstr 23226 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (EEG `  N ) Struct  <. 1 , ; 1 7 >. )
202, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  (EEG `  N ) Struct  <.
1 , ; 1 7 >. )
21 isstruct 14184 . . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  Fun  ( (EEG `  N )  \  { (/) } ) )
2320, 22syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  ( (EEG `  N )  \  { (/)
} ) )
24 structcnvcnv 14185 . . . . . . . . . 10  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2520, 24syl 16 . . . . . . . . 9  |-  ( ph  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2625funeqd 5439 . . . . . . . 8  |-  ( ph  ->  ( Fun  `' `' (EEG `  N )  <->  Fun  ( (EEG
`  N )  \  { (/) } ) ) )
2723, 26mpbird 232 . . . . . . 7  |-  ( ph  ->  Fun  `' `' (EEG
`  N ) )
28 opex 4556 . . . . . . . . . 10  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  _V
2928prid2 3984 . . . . . . . . 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 3523 . . . . . . . . 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 23225 . . . . . . . . 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 2530 . . . . . . 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 5701 . . . . . . . . 9  |-  ( EE
`  N )  e. 
_V
3635, 35mpt2ex 6650 . . . . . . . 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 14206 . . . . . 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 2494 . . . . 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 5693 . . . . . . . 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 5693 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( B `  i ) )
4542, 44oveq12d 6109 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( A `
 i )  -  ( B `  i ) ) )
4645oveq1d 6106 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
4746sumeq2dv 13180 . . . . 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 13165 . . . . . 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 6218 . . . 4  |-  ( ph  ->  ( A  .-  B
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
51 eqcom 2445 . . . . 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 5693 . . . . . . . 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 5693 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( D `  i ) )
5855, 57oveq12d 6109 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( C `
 i )  -  ( D `  i ) ) )
5958oveq1d 6106 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
6059sumeq2dv 13180 . . . . 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 13165 . . . . . 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 6218 . . . 4  |-  ( ph  ->  ( C  .-  D
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
64 eqcom 2445 . . . . 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 2457 . 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 964    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    \ cdif 3325    u. cun 3326    C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879   <.cop 3883   class class class wbr 4292   `'ccnv 4839   dom cdm 4840   Fun wfun 5412   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1c1 9283    <_ cle 9419    - cmin 9595   NNcn 10322   2c2 10371   7c7 10376  ;cdc 10755   ...cfz 11437   ^cexp 11865   sum_csu 13163   Struct cstr 14170   ndxcnx 14171   Basecbs 14174   distcds 14247  Itvcitv 22897  LineGclng 22898   EEcee 23134    Btwn cbtwn 23135  Cgrccgr 23136  EEGceeng 23223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-seq 11807  df-sum 13164  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-ds 14260  df-itv 22899  df-lng 22900  df-ee 23137  df-cgr 23139  df-eeng 23224
This theorem is referenced by:  eengtrkg  23231
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