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Theorem cnre2csqima 28332
Description: Image of a centered square by the canonical bijection from  ( RR  X.  RR ) to  CC. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Hypothesis
Ref Expression
cnre2csqima.1  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
Assertion
Ref Expression
cnre2csqima  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( (
( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  X.  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) )  -> 
( ( abs `  (
Re `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D  /\  ( abs `  ( Im
`  ( ( F `
 Y )  -  ( F `  X ) ) ) )  < 
D ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    D( x, y)    F( x, y)    X( x, y)    Y( x, y)

Proof of Theorem cnre2csqima
Dummy variables  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioossre 11638 . . 3  |-  ( ( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  C_  RR
2 ioossre 11638 . . 3  |-  ( ( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) )  C_  RR
3 xpinpreima2 28328 . . . 4  |-  ( ( ( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) 
C_  RR  /\  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) )  C_  RR )  ->  ( ( ( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  X.  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) ) "
( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" ( ( ( 2nd `  X )  -  D ) (,) ( ( 2nd `  X
)  +  D ) ) ) ) )
43eleq2d 2472 . . 3  |-  ( ( ( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) 
C_  RR  /\  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) )  C_  RR )  ->  ( Y  e.  ( ( ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X
)  +  D ) )  X.  ( ( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) )  <->  Y  e.  ( ( `' ( 1st  |`  ( RR  X.  RR ) ) "
( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" ( ( ( 2nd `  X )  -  D ) (,) ( ( 2nd `  X
)  +  D ) ) ) ) ) )
51, 2, 4mp2an 670 . 2  |-  ( Y  e.  ( ( ( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  X.  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) )  <->  Y  e.  ( ( `' ( 1st  |`  ( RR  X.  RR ) ) "
( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" ( ( ( 2nd `  X )  -  D ) (,) ( ( 2nd `  X
)  +  D ) ) ) ) )
6 elin 3625 . . 3  |-  ( Y  e.  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" ( ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X
)  +  D ) ) )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" ( ( ( 2nd `  X )  -  D ) (,) ( ( 2nd `  X
)  +  D ) ) ) )  <->  ( Y  e.  ( `' ( 1st  |`  ( RR  X.  RR ) ) " (
( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) ) )  /\  Y  e.  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
( ( ( 2nd `  X )  -  D
) (,) ( ( 2nd `  X )  +  D ) ) ) ) )
7 simpl 455 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  RR )
87recnd 9651 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
9 ax-icn 9580 . . . . . . . . . . . 12  |-  _i  e.  CC
109a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  _i  e.  CC )
11 simpr 459 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
1211recnd 9651 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1310, 12mulcld 9645 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
148, 13addcld 9644 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
15 reval 13086 . . . . . . . . 9  |-  ( ( x  +  ( _i  x.  y ) )  e.  CC  ->  (
Re `  ( x  +  ( _i  x.  y ) ) )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  ( x  +  (
_i  x.  y )
) ) )  / 
2 ) )
1614, 15syl 17 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) )
17 crre 13094 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  x )
1816, 17eqtr3d 2445 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
)  =  x )
1918mpt2eq3ia 6342 . . . . . 6  |-  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) )  =  ( x  e.  RR ,  y  e.  RR  |->  x )
2014adantl 464 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  ( _i  x.  y ) )  e.  CC )
21 cnre2csqima.1 . . . . . . . . 9  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
2221a1i 11 . . . . . . . 8  |-  ( T. 
->  F  =  (
x  e.  RR , 
y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) ) )
23 df-re 13080 . . . . . . . . 9  |-  Re  =  ( z  e.  CC  |->  ( ( z  +  ( * `  z
) )  /  2
) )
2423a1i 11 . . . . . . . 8  |-  ( T. 
->  Re  =  ( z  e.  CC  |->  ( ( z  +  ( * `
 z ) )  /  2 ) ) )
25 id 22 . . . . . . . . . 10  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  z  =  ( x  +  ( _i  x.  y
) ) )
26 fveq2 5848 . . . . . . . . . 10  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
* `  z )  =  ( * `  ( x  +  (
_i  x.  y )
) ) )
2725, 26oveq12d 6295 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  +  ( * `
 z ) )  =  ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) ) )
2827oveq1d 6292 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
( z  +  ( * `  z ) )  /  2 )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  ( x  +  (
_i  x.  y )
) ) )  / 
2 ) )
2920, 22, 24, 28fmpt2co 6866 . . . . . . 7  |-  ( T. 
->  ( Re  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) ) )
3029trud 1414 . . . . . 6  |-  ( Re  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
) )
31 df1stres 27952 . . . . . 6  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  x )
3219, 30, 313eqtr4ri 2442 . . . . 5  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( Re  o.  F )
3314rgen2a 2830 . . . . . 6  |-  A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC
3421fnmpt2 6851 . . . . . 6  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC  ->  F  Fn  ( RR 
X.  RR ) )
3533, 34ax-mp 5 . . . . 5  |-  F  Fn  ( RR  X.  RR )
36 fo1st 6803 . . . . . 6  |-  1st : _V -onto-> _V
37 fofn 5779 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3836, 37ax-mp 5 . . . . 5  |-  1st  Fn  _V
39 xp1st 6813 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
4021rnmpt2 6392 . . . . . . . 8  |-  ran  F  =  { z  |  E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y ) ) }
41 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( x  +  (
_i  x.  y )
) )
4214adantr 463 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
4341, 42eqeltrd 2490 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  z  e.  CC )
4443ex 432 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( z  =  ( x  +  ( _i  x.  y ) )  ->  z  e.  CC ) )
4544rexlimivv 2900 . . . . . . . . 9  |-  ( E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y
) )  ->  z  e.  CC )
4645abssi 3513 . . . . . . . 8  |-  { z  |  E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y ) ) }  C_  CC
4740, 46eqsstri 3471 . . . . . . 7  |-  ran  F  C_  CC
48 simpl 455 . . . . . . 7  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  z  e.  ran  F )
4947, 48sseldi 3439 . . . . . 6  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  z  e.  CC )
50 simpr 459 . . . . . . 7  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  u  e.  ran  F )
5147, 50sseldi 3439 . . . . . 6  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  u  e.  CC )
5249, 51resubd 13196 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Re `  ( z  -  u
) )  =  ( ( Re `  z
)  -  ( Re
`  u ) ) )
5332, 35, 38, 39, 52cnre2csqlem 28331 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( 1st  |`  ( RR  X.  RR ) ) "
( ( ( 1st `  X )  -  D
) (,) ( ( 1st `  X )  +  D ) ) )  ->  ( abs `  ( Re `  (
( F `  Y
)  -  ( F `
 X ) ) ) )  <  D
) )
54 imval 13087 . . . . . . . . 9  |-  ( ( x  +  ( _i  x.  y ) )  e.  CC  ->  (
Im `  ( x  +  ( _i  x.  y ) ) )  =  ( Re `  ( ( x  +  ( _i  x.  y
) )  /  _i ) ) )
5514, 54syl 17 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )
56 crim 13095 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  y )
5755, 56eqtr3d 2445 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) )  =  y )
5857mpt2eq3ia 6342 . . . . . 6  |-  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )  =  ( x  e.  RR ,  y  e.  RR  |->  y )
59 df-im 13081 . . . . . . . . 9  |-  Im  =  ( z  e.  CC  |->  ( Re `  ( z  /  _i ) ) )
6059a1i 11 . . . . . . . 8  |-  ( T. 
->  Im  =  ( z  e.  CC  |->  ( Re
`  ( z  /  _i ) ) ) )
61 oveq1 6284 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  /  _i )  =  ( ( x  +  ( _i  x.  y ) )  /  _i ) )
6261fveq2d 5852 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
Re `  ( z  /  _i ) )  =  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) ) )
6320, 22, 60, 62fmpt2co 6866 . . . . . . 7  |-  ( T. 
->  ( Im  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) ) )
6463trud 1414 . . . . . 6  |-  ( Im  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re `  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )
65 df2ndres 27953 . . . . . 6  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  y )
6658, 64, 653eqtr4ri 2442 . . . . 5  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( Im  o.  F )
67 fo2nd 6804 . . . . . 6  |-  2nd : _V -onto-> _V
68 fofn 5779 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
6967, 68ax-mp 5 . . . . 5  |-  2nd  Fn  _V
70 xp2nd 6814 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
7149, 51imsubd 13197 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Im `  ( z  -  u
) )  =  ( ( Im `  z
)  -  ( Im
`  u ) ) )
7266, 35, 69, 70, 71cnre2csqlem 28331 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
( ( ( 2nd `  X )  -  D
) (,) ( ( 2nd `  X )  +  D ) ) )  ->  ( abs `  ( Im `  (
( F `  Y
)  -  ( F `
 X ) ) ) )  <  D
) )
7353, 72anim12d 561 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( Y  e.  ( `' ( 1st  |`  ( RR  X.  RR ) )
" ( ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X
)  +  D ) ) )  /\  Y  e.  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) ) )  ->  ( ( abs `  ( Re `  (
( F `  Y
)  -  ( F `
 X ) ) ) )  <  D  /\  ( abs `  (
Im `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D ) ) )
746, 73syl5bi 217 . 2  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" ( ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X
)  +  D ) ) )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" ( ( ( 2nd `  X )  -  D ) (,) ( ( 2nd `  X
)  +  D ) ) ) )  -> 
( ( abs `  (
Re `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D  /\  ( abs `  ( Im
`  ( ( F `
 Y )  -  ( F `  X ) ) ) )  < 
D ) ) )
755, 74syl5bi 217 1  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( (
( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  X.  (
( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) ) )  -> 
( ( abs `  (
Re `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D  /\  ( abs `  ( Im
`  ( ( F `
 Y )  -  ( F `  X ) ) ) )  < 
D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   T. wtru 1406    e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754   _Vcvv 3058    i^i cin 3412    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820   `'ccnv 4821   ran crn 4823    |` cres 4824   "cima 4825    o. ccom 4826    Fn wfn 5563   -onto->wfo 5566   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6781   2ndc2nd 6782   CCcc 9519   RRcr 9520   _ici 9523    + caddc 9524    x. cmul 9526    < clt 9657    - cmin 9840    / cdiv 10246   2c2 10625   RR+crp 11264   (,)cioo 11581   *ccj 13076   Recre 13077   Imcim 13078   abscabs 13214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ioo 11585  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216
This theorem is referenced by:  tpr2rico  28333
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