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Theorem cnpart 13039
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 2665 . . . . . 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 10626 . . . . . . . 8  |-  0  <_  0
42, 3syl6eqbr 4484 . . . . . . 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 9552 . . . . . . . . . . . 12  |-  _i  e.  CC
9 mulcl 9577 . . . . . . . . . . . 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 12918 . . . . . . . . . . 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 12907 . . . . . . . . . . . . 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 9993 . . . . . . . . . . . . . . . . 17  |-  _i  =/=  0
16 divrec2 10225 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  A
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
178, 15, 16mp3an23 1316 . . . . . . . . . . . . . . . 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 12236 . . . . . . . . . . . . . . . 16  |-  ( 1  /  _i )  = 
-u _i
2019oveq1i 6295 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  _i )  x.  ( _i  x.  A ) )  =  ( -u _i  x.  ( _i  x.  A
) )
2118, 20syl6eq 2524 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( -u _i  x.  ( _i  x.  A
) ) )
22 divcan3 10232 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  A )
238, 15, 22mp3an23 1316 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  A )
2421, 23eqtr3d 2510 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2524fveq2d 5870 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Re `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( Re `  A
) )
2614, 25eqtrd 2508 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  A
) )
2726eqeq1d 2469 . . . . . . . . . 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 10192 . . . . . . . . 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 11250 . . . . . . 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 2665 . . . . 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 4483 . . . . 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 2714 . . . . . . . . 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 11227 . . . . . . . 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 9597 . . . . . . . 8  |-  0  e.  RR
52 recl 12909 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
53 ltlen 9687 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
54 ltnle 9665 . . . . . . . . 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 9883 . . . . . . . . . 10  |-  ( -u ( _i  x.  A
)  e.  RR  ->  -u -u ( _i  x.  A
)  e.  RR )
6110negnegd 9922 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  -u -u (
_i  x.  A )  =  ( _i  x.  A ) )
6261eleq1d 2536 . . . . . . . . . . 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 11227 . . . . . . . 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 2785 . 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 12924 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  -u A )  =  -u ( Re `  A ) )
7473breq2d 4459 . . . . . 6  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  0  <_  -u ( Re `  A ) ) )
7552le0neg1d 10125 . . . . . 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 9995 . . . . . . 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 2805 . . . . . 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 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494   _ici 9495    x. cmul 9498    < clt 9629    <_ cle 9630   -ucneg 9807    / cdiv 10207   RR+crp 11221   Recre 12896   Imcim 12897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-2 10595  df-rp 11222  df-cj 12898  df-re 12899  df-im 12900
This theorem is referenced by:  sqrmo  13051
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