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Theorem cnre2csqima 27515
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 11575 . . 3  |-  ( ( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  C_  RR
2 ioossre 11575 . . 3  |-  ( ( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) )  C_  RR
3 xpinpreima2 27511 . . . 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 2530 . . 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 672 . 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 3680 . . 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 457 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  RR )
87recnd 9611 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
9 ax-icn 9540 . . . . . . . . . . . 12  |-  _i  e.  CC
109a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  _i  e.  CC )
11 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
1211recnd 9611 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1310, 12mulcld 9605 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
148, 13addcld 9604 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
15 reval 12889 . . . . . . . . 9  |-  ( ( x  +  ( _i  x.  y ) )  e.  CC  ->  (
Re `  ( x  +  ( _i  x.  y ) ) )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  ( x  +  (
_i  x.  y )
) ) )  / 
2 ) )
1614, 15syl 16 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) )
17 crre 12897 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  x )
1816, 17eqtr3d 2503 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
)  =  x )
1918mpt2eq3ia 6337 . . . . . 6  |-  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) )  =  ( x  e.  RR ,  y  e.  RR  |->  x )
2014adantl 466 . . . . . . . 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 12883 . . . . . . . . 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 5857 . . . . . . . . . 10  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
* `  z )  =  ( * `  ( x  +  (
_i  x.  y )
) ) )
2725, 26oveq12d 6293 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  +  ( * `
 z ) )  =  ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) ) )
2827oveq1d 6290 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
( z  +  ( * `  z ) )  /  2 )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  ( x  +  (
_i  x.  y )
) ) )  / 
2 ) )
2920, 22, 24, 28fmpt2co 6856 . . . . . . 7  |-  ( T. 
->  ( Re  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) ) )
3029trud 1383 . . . . . 6  |-  ( Re  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
) )
31 df1stres 27180 . . . . . 6  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  x )
3219, 30, 313eqtr4ri 2500 . . . . 5  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( Re  o.  F )
3314rgen2a 2884 . . . . . 6  |-  A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC
3421fnmpt2 6842 . . . . . 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 6794 . . . . . 6  |-  1st : _V -onto-> _V
37 fofn 5788 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3836, 37ax-mp 5 . . . . 5  |-  1st  Fn  _V
39 xp1st 6804 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
4021rnmpt2 6387 . . . . . . . 8  |-  ran  F  =  { z  |  E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y ) ) }
41 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( x  +  (
_i  x.  y )
) )
4214adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
4341, 42eqeltrd 2548 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  =  ( x  +  ( _i  x.  y ) ) )  ->  z  e.  CC )
4443ex 434 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( z  =  ( x  +  ( _i  x.  y ) )  ->  z  e.  CC ) )
4544rexlimivv 2953 . . . . . . . . 9  |-  ( E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y
) )  ->  z  e.  CC )
4645abssi 3568 . . . . . . . 8  |-  { z  |  E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y ) ) }  C_  CC
4740, 46eqsstri 3527 . . . . . . 7  |-  ran  F  C_  CC
48 simpl 457 . . . . . . 7  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  z  e.  ran  F )
4947, 48sseldi 3495 . . . . . 6  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  z  e.  CC )
50 simpr 461 . . . . . . 7  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  u  e.  ran  F )
5147, 50sseldi 3495 . . . . . 6  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  u  e.  CC )
5249, 51resubd 12999 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Re `  ( z  -  u
) )  =  ( ( Re `  z
)  -  ( Re
`  u ) ) )
5332, 35, 38, 39, 52cnre2csqlem 27514 . . . 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 12890 . . . . . . . . 9  |-  ( ( x  +  ( _i  x.  y ) )  e.  CC  ->  (
Im `  ( x  +  ( _i  x.  y ) ) )  =  ( Re `  ( ( x  +  ( _i  x.  y
) )  /  _i ) ) )
5514, 54syl 16 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )
56 crim 12898 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  y )
5755, 56eqtr3d 2503 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) )  =  y )
5857mpt2eq3ia 6337 . . . . . 6  |-  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )  =  ( x  e.  RR ,  y  e.  RR  |->  y )
59 df-im 12884 . . . . . . . . 9  |-  Im  =  ( z  e.  CC  |->  ( Re `  ( z  /  _i ) ) )
6059a1i 11 . . . . . . . 8  |-  ( T. 
->  Im  =  ( z  e.  CC  |->  ( Re
`  ( z  /  _i ) ) ) )
61 oveq1 6282 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  /  _i )  =  ( ( x  +  ( _i  x.  y ) )  /  _i ) )
6261fveq2d 5861 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
Re `  ( z  /  _i ) )  =  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) ) )
6320, 22, 60, 62fmpt2co 6856 . . . . . . 7  |-  ( T. 
->  ( Im  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) ) )
6463trud 1383 . . . . . 6  |-  ( Im  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re `  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )
65 df2ndres 27181 . . . . . 6  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  y )
6658, 64, 653eqtr4ri 2500 . . . . 5  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( Im  o.  F )
67 fo2nd 6795 . . . . . 6  |-  2nd : _V -onto-> _V
68 fofn 5788 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
6967, 68ax-mp 5 . . . . 5  |-  2nd  Fn  _V
70 xp2nd 6805 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
7149, 51imsubd 13000 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Im `  ( z  -  u
) )  =  ( ( Im `  z
)  -  ( Im
`  u ) ) )
7266, 35, 69, 70, 71cnre2csqlem 27514 . . . 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 563 . . 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 369    /\ w3a 968    = wceq 1374   T. wtru 1375    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   ran crn 4993    |` cres 4994   "cima 4995    o. ccom 4996    Fn wfn 5574   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   CCcc 9479   RRcr 9480   _ici 9483    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9794    / cdiv 10195   2c2 10574   RR+crp 11209   (,)cioo 11518   *ccj 12879   Recre 12880   Imcim 12881   abscabs 13017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ioo 11522  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019
This theorem is referenced by:  tpr2rico  27516
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