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Theorem cnre2csqima 26479
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 11461 . . 3  |-  ( ( ( 1st `  X
)  -  D ) (,) ( ( 1st `  X )  +  D
) )  C_  RR
2 ioossre 11461 . . 3  |-  ( ( ( 2nd `  X
)  -  D ) (,) ( ( 2nd `  X )  +  D
) )  C_  RR
3 xpinpreima2 26475 . . . 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 2521 . . 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 3640 . . 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 9516 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
9 ax-icn 9445 . . . . . . . . . . . 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 9516 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1310, 12mulcld 9510 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
148, 13addcld 9509 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
15 reval 12706 . . . . . . . . 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 12714 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  x )
1816, 17eqtr3d 2494 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
)  =  x )
1918mpt2eq3ia 6253 . . . . . 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 12700 . . . . . . . . 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 5792 . . . . . . . . . 10  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
* `  z )  =  ( * `  ( x  +  (
_i  x.  y )
) ) )
2725, 26oveq12d 6211 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  +  ( * `
 z ) )  =  ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) ) )
2827oveq1d 6208 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
( z  +  ( * `  z ) )  /  2 )  =  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  ( x  +  (
_i  x.  y )
) ) )  / 
2 ) )
2920, 22, 24, 28fmpt2co 6759 . . . . . . 7  |-  ( T. 
->  ( Re  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `
 ( x  +  ( _i  x.  y
) ) ) )  /  2 ) ) )
3029trud 1379 . . . . . 6  |-  ( Re  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( ( ( x  +  ( _i  x.  y ) )  +  ( * `  (
x  +  ( _i  x.  y ) ) ) )  /  2
) )
31 df1stres 26143 . . . . . 6  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  x )
3219, 30, 313eqtr4ri 2491 . . . . 5  |-  ( 1st  |`  ( RR  X.  RR ) )  =  ( Re  o.  F )
3314rgen2a 2893 . . . . . 6  |-  A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC
3421fnmpt2 6745 . . . . . 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 6699 . . . . . 6  |-  1st : _V -onto-> _V
37 fofn 5723 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3836, 37ax-mp 5 . . . . 5  |-  1st  Fn  _V
39 xp1st 6709 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
4021rnmpt2 6303 . . . . . . . 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 2539 . . . . . . . . . . 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 2945 . . . . . . . . 9  |-  ( E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y
) )  ->  z  e.  CC )
4645abssi 3528 . . . . . . . 8  |-  { z  |  E. x  e.  RR  E. y  e.  RR  z  =  ( x  +  ( _i  x.  y ) ) }  C_  CC
4740, 46eqsstri 3487 . . . . . . 7  |-  ran  F  C_  CC
48 simpl 457 . . . . . . 7  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  z  e.  ran  F )
4947, 48sseldi 3455 . . . . . 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 3455 . . . . . 6  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  u  e.  CC )
5249, 51resubd 12816 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Re `  ( z  -  u
) )  =  ( ( Re `  z
)  -  ( Re
`  u ) ) )
5332, 35, 38, 39, 52cnre2csqlem 26478 . . . 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 12707 . . . . . . . . 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 12715 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  y )
5755, 56eqtr3d 2494 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) )  =  y )
5857mpt2eq3ia 6253 . . . . . 6  |-  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )  =  ( x  e.  RR ,  y  e.  RR  |->  y )
59 df-im 12701 . . . . . . . . 9  |-  Im  =  ( z  e.  CC  |->  ( Re `  ( z  /  _i ) ) )
6059a1i 11 . . . . . . . 8  |-  ( T. 
->  Im  =  ( z  e.  CC  |->  ( Re
`  ( z  /  _i ) ) ) )
61 oveq1 6200 . . . . . . . . 9  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
z  /  _i )  =  ( ( x  +  ( _i  x.  y ) )  /  _i ) )
6261fveq2d 5796 . . . . . . . 8  |-  ( z  =  ( x  +  ( _i  x.  y
) )  ->  (
Re `  ( z  /  _i ) )  =  ( Re `  (
( x  +  ( _i  x.  y ) )  /  _i ) ) )
6320, 22, 60, 62fmpt2co 6759 . . . . . . 7  |-  ( T. 
->  ( Im  o.  F
)  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re
`  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) ) )
6463trud 1379 . . . . . 6  |-  ( Im  o.  F )  =  ( x  e.  RR ,  y  e.  RR  |->  ( Re `  ( ( x  +  ( _i  x.  y ) )  /  _i ) ) )
65 df2ndres 26144 . . . . . 6  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  y )
6658, 64, 653eqtr4ri 2491 . . . . 5  |-  ( 2nd  |`  ( RR  X.  RR ) )  =  ( Im  o.  F )
67 fo2nd 6700 . . . . . 6  |-  2nd : _V -onto-> _V
68 fofn 5723 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
6967, 68ax-mp 5 . . . . 5  |-  2nd  Fn  _V
70 xp2nd 6710 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
7149, 51imsubd 12817 . . . . 5  |-  ( ( z  e.  ran  F  /\  u  e.  ran  F )  ->  ( Im `  ( z  -  u
) )  =  ( ( Im `  z
)  -  ( Im
`  u ) ) )
7266, 35, 69, 70, 71cnre2csqlem 26478 . . . 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 965    = wceq 1370   T. wtru 1371    e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796   _Vcvv 3071    i^i cin 3428    C_ wss 3429   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939   `'ccnv 4940   ran crn 4942    |` cres 4943   "cima 4944    o. ccom 4945    Fn wfn 5514   -onto->wfo 5517   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679   CCcc 9384   RRcr 9385   _ici 9388    + caddc 9389    x. cmul 9391    < clt 9522    - cmin 9699    / cdiv 10097   2c2 10475   RR+crp 11095   (,)cioo 11404   *ccj 12696   Recre 12697   Imcim 12698   abscabs 12834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-ioo 11408  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836
This theorem is referenced by:  tpr2rico  26480
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