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Theorem cnpart 13222
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 2601 . . . . . 6  |-  ( -u ( _i  x.  A
)  e/  RR+  <->  -.  -u (
_i  x.  A )  e.  RR+ )
2 simpr 459 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  =  0 )
3 0le0 10666 . . . . . . . 8  |-  0  <_  0
42, 3syl6eqbr 4432 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  <_  0
)
54biantrurd 506 . . . . . 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 328 . . . 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 9581 . . . . . . . . . . . 12  |-  _i  e.  CC
9 mulcl 9606 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
108, 9mpan 668 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
11 reim0b 13101 . . . . . . . . . . 11  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
1210, 11syl 17 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
13 imre 13090 . . . . . . . . . . . . 13  |-  ( ( _i  x.  A )  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
1410, 13syl 17 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
15 ine0 10033 . . . . . . . . . . . . . . . . 17  |-  _i  =/=  0
16 divrec2 10265 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  A
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
178, 15, 16mp3an23 1318 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
1810, 17syl 17 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
19 irec 12312 . . . . . . . . . . . . . . . 16  |-  ( 1  /  _i )  = 
-u _i
2019oveq1i 6288 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  _i )  x.  ( _i  x.  A ) )  =  ( -u _i  x.  ( _i  x.  A
) )
2118, 20syl6eq 2459 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( -u _i  x.  ( _i  x.  A
) ) )
22 divcan3 10272 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  A )
238, 15, 22mp3an23 1318 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  A )
2421, 23eqtr3d 2445 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2524fveq2d 5853 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Re `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( Re `  A
) )
2614, 25eqtrd 2443 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  A
) )
2726eqeq1d 2404 . . . . . . . . . 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 483 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =  0 )  -> 
( _i  x.  A
)  e.  RR )
3029adantlr 713 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  e.  RR )
31 mulne0 10232 . . . . . . . . 9  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( _i  x.  A
)  =/=  0 )
328, 15, 31mpanl12 680 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  A
)  =/=  0 )
3332adantr 463 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  =/=  0
)
34 rpneg 11295 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  RR  /\  ( _i  x.  A
)  =/=  0 )  ->  ( ( _i  x.  A )  e.  RR+ 
<->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3530, 33, 34syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
_i  x.  A )  e.  RR+  <->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3635con2bid 327 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  -.  ( _i  x.  A )  e.  RR+ ) )
37 df-nel 2601 . . . . 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 4431 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  0  <_  ( Re `  A ) )
4039biantrurd 506 . . . 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 463 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =  0 ) )
4342necon3bbid 2650 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =/=  0
) )
4443biimpar 483 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR )
45 rpre 11271 . . . . . . . 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 505 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ ) ) )
49 simpr 459 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( Re `  A )  =/=  0
)
5049biantrud 505 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
51 0re 9626 . . . . . . . 8  |-  0  e.  RR
52 recl 13092 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
53 ltlen 9717 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
54 ltnle 9695 . . . . . . . . 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 661 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  A )  /\  ( Re `  A
)  =/=  0 )  <->  -.  ( Re `  A
)  <_  0 ) )
5756ad2antrr 724 . . . . . 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 9918 . . . . . . . . . 10  |-  ( -u ( _i  x.  A
)  e.  RR  ->  -u -u ( _i  x.  A
)  e.  RR )
6110negnegd 9958 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  -u -u (
_i  x.  A )  =  ( _i  x.  A ) )
6261eleq1d 2471 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( -u -u ( _i  x.  A
)  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6362ad2antrr 724 . . . . . . . . . 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 11271 . . . . . . . 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 505 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
Re `  A )  <_  0  <->  ( ( Re
`  A )  <_ 
0  /\  -u ( _i  x.  A )  e/  RR+ ) ) )
7069notbid 292 . . . 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 2721 . 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 13107 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  -u A )  =  -u ( Re `  A ) )
7473breq2d 4407 . . . . . 6  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  0  <_  -u ( Re `  A ) ) )
7552le0neg1d 10164 . . . . . 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 10035 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
788, 77mpan 668 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
79 neleq1 2742 . . . . . 6  |-  ( ( _i  x.  -u A
)  =  -u (
_i  x.  A )  ->  ( ( _i  x.  -u A )  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8078, 79syl 17 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8176, 80anbi12d 709 . . . 4  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  ( ( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
8281notbid 292 . . 3  |-  ( A  e.  CC  ->  ( -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )
) )
8382adantr 463 . 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598    e/ wnel 2599   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523   _ici 9524    x. cmul 9527    < clt 9658    <_ cle 9659   -ucneg 9842    / cdiv 10247   RR+crp 11265   Recre 13079   Imcim 13080
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 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-rp 11266  df-cj 13081  df-re 13082  df-im 13083
This theorem is referenced by:  sqrmo  13234
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