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Theorem cnpart 12050
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 2604 . . . . . 6  |-  ( -u ( _i  x.  A
)  e/  RR+  <->  -.  -u (
_i  x.  A )  e.  RR+ )
2 simpr 449 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  =  0 )
3 0le0 10086 . . . . . . . 8  |-  0  <_  0
42, 3syl6eqbr 4252 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  <_  0
)
54biantrurd 496 . . . . . 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 252 . . . . 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 322 . . . 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 9054 . . . . . . . . . . . 12  |-  _i  e.  CC
9 mulcl 9079 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
108, 9mpan 653 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
11 reim0b 11929 . . . . . . . . . . 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 11918 . . . . . . . . . . . . 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 9474 . . . . . . . . . . . . . . . . 17  |-  _i  =/=  0
16 divrec2 9700 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  A
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
178, 15, 16mp3an23 1272 . . . . . . . . . . . . . . . 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 11485 . . . . . . . . . . . . . . . 16  |-  ( 1  /  _i )  = 
-u _i
2019oveq1i 6094 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  _i )  x.  ( _i  x.  A ) )  =  ( -u _i  x.  ( _i  x.  A
) )
2118, 20syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( -u _i  x.  ( _i  x.  A
) ) )
22 divcan3 9707 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  A )
238, 15, 22mp3an23 1272 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  A )
2421, 23eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2524fveq2d 5735 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Re `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( Re `  A
) )
2614, 25eqtrd 2470 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  A
) )
2726eqeq1d 2446 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( Im `  (
_i  x.  A )
)  =  0  <->  (
Re `  A )  =  0 ) )
2812, 27bitrd 246 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Re `  A )  =  0 ) )
2928biimpar 473 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =  0 )  -> 
( _i  x.  A
)  e.  RR )
3029adantlr 697 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  e.  RR )
31 mulne0 9669 . . . . . . . . 9  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( _i  x.  A
)  =/=  0 )
328, 15, 31mpanl12 665 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  A
)  =/=  0 )
3332adantr 453 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  =/=  0
)
34 rpneg 10646 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  RR  /\  ( _i  x.  A
)  =/=  0 )  ->  ( ( _i  x.  A )  e.  RR+ 
<->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3530, 33, 34syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
_i  x.  A )  e.  RR+  <->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3635con2bid 321 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  -.  ( _i  x.  A )  e.  RR+ ) )
37 df-nel 2604 . . . . 5  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
3836, 37syl6bbr 256 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  ( _i  x.  A )  e/  RR+ )
)
393, 2syl5breqr 4251 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  0  <_  ( Re `  A ) )
4039biantrurd 496 . . . 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 273 . . 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 453 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =  0 ) )
4342necon3bbid 2637 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =/=  0
) )
4443biimpar 473 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR )
45 rpre 10623 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  RR+  ->  ( _i  x.  A )  e.  RR )
4644, 45nsyl 116 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR+ )
4746, 37sylibr 205 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( _i  x.  A )  e/  RR+ )
4847biantrud 495 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ ) ) )
49 simpr 449 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( Re `  A )  =/=  0
)
5049biantrud 495 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
51 0re 9096 . . . . . . . 8  |-  0  e.  RR
52 recl 11920 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
53 ltlen 9180 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
54 ltnle 9160 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  -.  ( Re `  A
)  <_  0 ) )
5553, 54bitr3d 248 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( Re `  A )  =/=  0
)  <->  -.  ( Re `  A )  <_  0
) )
5651, 52, 55sylancr 646 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  A )  /\  ( Re `  A
)  =/=  0 )  <->  -.  ( Re `  A
)  <_  0 ) )
5756ad2antrr 708 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 )  <->  -.  (
Re `  A )  <_  0 ) )
5850, 57bitrd 246 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  -.  (
Re `  A )  <_  0 ) )
5948, 58bitr3d 248 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ )  <->  -.  (
Re `  A )  <_  0 ) )
60 renegcl 9369 . . . . . . . . . 10  |-  ( -u ( _i  x.  A
)  e.  RR  ->  -u -u ( _i  x.  A
)  e.  RR )
6110negnegd 9407 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  -u -u (
_i  x.  A )  =  ( _i  x.  A ) )
6261eleq1d 2504 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( -u -u ( _i  x.  A
)  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6362ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u -u (
_i  x.  A )  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6460, 63syl5ib 212 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u (
_i  x.  A )  e.  RR  ->  ( _i  x.  A )  e.  RR ) )
6544, 64mtod 171 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR )
66 rpre 10623 . . . . . . . 8  |-  ( -u ( _i  x.  A
)  e.  RR+  ->  -u ( _i  x.  A
)  e.  RR )
6765, 66nsyl 116 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR+ )
6867, 1sylibr 205 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -u ( _i  x.  A )  e/  RR+ )
6968biantrud 495 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
Re `  A )  <_  0  <->  ( ( Re
`  A )  <_ 
0  /\  -u ( _i  x.  A )  e/  RR+ ) ) )
7069notbid 287 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -.  ( Re `  A )  <_  0  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
7159, 70bitrd 246 . . 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 2684 . 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 11935 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  -u A )  =  -u ( Re `  A ) )
7473breq2d 4227 . . . . . 6  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  0  <_  -u ( Re `  A ) ) )
7552le0neg1d 9603 . . . . . 6  |-  ( A  e.  CC  ->  (
( Re `  A
)  <_  0  <->  0  <_  -u ( Re `  A ) ) )
7674, 75bitr4d 249 . . . . 5  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  ( Re `  A )  <_  0
) )
77 mulneg2 9476 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
788, 77mpan 653 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
79 neleq1 2701 . . . . . 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 693 . . . 4  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  ( ( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
8281notbid 287 . . 3  |-  ( A  e.  CC  ->  ( -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )
) )
8382adantr 453 . 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 249 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    e/ wnel 2602   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   _ici 8997    x. cmul 9000    < clt 9125    <_ cle 9126   -ucneg 9297    / cdiv 9682   RR+crp 10617   Recre 11907   Imcim 11908
This theorem is referenced by:  sqrmo  12062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-rp 10618  df-cj 11909  df-re 11910  df-im 11911
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