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Theorem cnpart 12721
Description: The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map  x 
|->  -u x). (Contributed by Mario Carneiro, 8-Jul-2013.)
Assertion
Ref Expression
cnpart  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( _i  x.  A )  e/  RR+ )  <->  -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )
) )

Proof of Theorem cnpart
StepHypRef Expression
1 df-nel 2603 . . . . . 6  |-  ( -u ( _i  x.  A
)  e/  RR+  <->  -.  -u (
_i  x.  A )  e.  RR+ )
2 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  =  0 )
3 0le0 10403 . . . . . . . 8  |-  0  <_  0
42, 3syl6eqbr 4322 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  <_  0
)
54biantrurd 508 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e/  RR+  <->  ( ( Re
`  A )  <_ 
0  /\  -u ( _i  x.  A )  e/  RR+ ) ) )
61, 5syl5bbr 259 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -.  -u ( _i  x.  A
)  e.  RR+  <->  ( (
Re `  A )  <_  0  /\  -u (
_i  x.  A )  e/  RR+ ) ) )
76con1bid 330 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )  <->  -u ( _i  x.  A
)  e.  RR+ )
)
8 ax-icn 9333 . . . . . . . . . . . 12  |-  _i  e.  CC
9 mulcl 9358 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
108, 9mpan 670 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
11 reim0b 12600 . . . . . . . . . . 11  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
13 imre 12589 . . . . . . . . . . . . 13  |-  ( ( _i  x.  A )  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
1410, 13syl 16 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
15 ine0 9772 . . . . . . . . . . . . . . . . 17  |-  _i  =/=  0
16 divrec2 10003 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  A
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
178, 15, 16mp3an23 1306 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
1810, 17syl 16 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
19 irec 11957 . . . . . . . . . . . . . . . 16  |-  ( 1  /  _i )  = 
-u _i
2019oveq1i 6096 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  _i )  x.  ( _i  x.  A ) )  =  ( -u _i  x.  ( _i  x.  A
) )
2118, 20syl6eq 2485 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( -u _i  x.  ( _i  x.  A
) ) )
22 divcan3 10010 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  A )
238, 15, 22mp3an23 1306 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  A )
2421, 23eqtr3d 2471 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2524fveq2d 5688 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Re `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( Re `  A
) )
2614, 25eqtrd 2469 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  A
) )
2726eqeq1d 2445 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( Im `  (
_i  x.  A )
)  =  0  <->  (
Re `  A )  =  0 ) )
2812, 27bitrd 253 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Re `  A )  =  0 ) )
2928biimpar 485 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =  0 )  -> 
( _i  x.  A
)  e.  RR )
3029adantlr 714 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  e.  RR )
31 mulne0 9970 . . . . . . . . 9  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( _i  x.  A
)  =/=  0 )
328, 15, 31mpanl12 682 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  A
)  =/=  0 )
3332adantr 465 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  =/=  0
)
34 rpneg 11012 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  RR  /\  ( _i  x.  A
)  =/=  0 )  ->  ( ( _i  x.  A )  e.  RR+ 
<->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3530, 33, 34syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
_i  x.  A )  e.  RR+  <->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3635con2bid 329 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  -.  ( _i  x.  A )  e.  RR+ ) )
37 df-nel 2603 . . . . 5  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
3836, 37syl6bbr 263 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  ( _i  x.  A )  e/  RR+ )
)
393, 2syl5breqr 4321 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  0  <_  ( Re `  A ) )
4039biantrurd 508 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
_i  x.  A )  e/  RR+  <->  ( 0  <_ 
( Re `  A
)  /\  ( _i  x.  A )  e/  RR+ )
) )
417, 38, 403bitrrd 280 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ )  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
4228adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =  0 ) )
4342necon3bbid 2636 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =/=  0
) )
4443biimpar 485 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR )
45 rpre 10989 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  RR+  ->  ( _i  x.  A )  e.  RR )
4644, 45nsyl 121 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR+ )
4746, 37sylibr 212 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( _i  x.  A )  e/  RR+ )
4847biantrud 507 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ ) ) )
49 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( Re `  A )  =/=  0
)
5049biantrud 507 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
51 0re 9378 . . . . . . . 8  |-  0  e.  RR
52 recl 12591 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
53 ltlen 9468 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
54 ltnle 9446 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  -.  ( Re `  A
)  <_  0 ) )
5553, 54bitr3d 255 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( Re `  A )  =/=  0
)  <->  -.  ( Re `  A )  <_  0
) )
5651, 52, 55sylancr 663 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  A )  /\  ( Re `  A
)  =/=  0 )  <->  -.  ( Re `  A
)  <_  0 ) )
5756ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 )  <->  -.  (
Re `  A )  <_  0 ) )
5850, 57bitrd 253 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  -.  (
Re `  A )  <_  0 ) )
5948, 58bitr3d 255 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ )  <->  -.  (
Re `  A )  <_  0 ) )
60 renegcl 9664 . . . . . . . . . 10  |-  ( -u ( _i  x.  A
)  e.  RR  ->  -u -u ( _i  x.  A
)  e.  RR )
6110negnegd 9702 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  -u -u (
_i  x.  A )  =  ( _i  x.  A ) )
6261eleq1d 2503 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( -u -u ( _i  x.  A
)  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6362ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u -u (
_i  x.  A )  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6460, 63syl5ib 219 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u (
_i  x.  A )  e.  RR  ->  ( _i  x.  A )  e.  RR ) )
6544, 64mtod 177 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR )
66 rpre 10989 . . . . . . . 8  |-  ( -u ( _i  x.  A
)  e.  RR+  ->  -u ( _i  x.  A
)  e.  RR )
6765, 66nsyl 121 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR+ )
6867, 1sylibr 212 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -u ( _i  x.  A )  e/  RR+ )
6968biantrud 507 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
Re `  A )  <_  0  <->  ( ( Re
`  A )  <_ 
0  /\  -u ( _i  x.  A )  e/  RR+ ) ) )
7069notbid 294 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -.  ( Re `  A )  <_  0  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
7159, 70bitrd 253 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ )  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
7241, 71pm2.61dane 2683 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( _i  x.  A )  e/  RR+ )  <->  -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )
) )
73 reneg 12606 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  -u A )  =  -u ( Re `  A ) )
7473breq2d 4297 . . . . . 6  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  0  <_  -u ( Re `  A ) ) )
7552le0neg1d 9903 . . . . . 6  |-  ( A  e.  CC  ->  (
( Re `  A
)  <_  0  <->  0  <_  -u ( Re `  A ) ) )
7674, 75bitr4d 256 . . . . 5  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  ( Re `  A )  <_  0
) )
77 mulneg2 9774 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
788, 77mpan 670 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
79 neleq1 2703 . . . . . 6  |-  ( ( _i  x.  -u A
)  =  -u (
_i  x.  A )  ->  ( ( _i  x.  -u A )  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8078, 79syl 16 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8176, 80anbi12d 710 . . . 4  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  ( ( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
8281notbid 294 . . 3  |-  ( A  e.  CC  ->  ( -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )
) )
8382adantr 465 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( 0  <_  ( Re `  -u A )  /\  (
_i  x.  -u A )  e/  RR+ )  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
8472, 83bitr4d 256 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( _i  x.  A )  e/  RR+ )  <->  -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2600    e/ wnel 2601   class class class wbr 4285   ` cfv 5411  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   _ici 9276    x. cmul 9279    < clt 9410    <_ cle 9411   -ucneg 9588    / cdiv 9985   RR+crp 10983   Recre 12578   Imcim 12579
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 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-po 4633  df-so 4634  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-rp 10984  df-cj 12580  df-re 12581  df-im 12582
This theorem is referenced by:  sqrmo  12733
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