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Theorem cnpart 11932
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 2532 . . . . . 6  |-  ( -u ( _i  x.  A
)  e/  RR+  <->  -.  -u (
_i  x.  A )  e.  RR+ )
2 simpr 447 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  =  0 )
3 0le0 9974 . . . . . . . 8  |-  0  <_  0
42, 3syl6eqbr 4162 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( Re `  A )  <_  0
)
54biantrurd 494 . . . . . 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 250 . . . . 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 320 . . . 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 8943 . . . . . . . . . . . 12  |-  _i  e.  CC
9 mulcl 8968 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
108, 9mpan 651 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
11 reim0b 11811 . . . . . . . . . . 11  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
1210, 11syl 15 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Im `  ( _i  x.  A
) )  =  0 ) )
13 imre 11800 . . . . . . . . . . . . 13  |-  ( ( _i  x.  A )  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
1410, 13syl 15 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  ( -u _i  x.  ( _i  x.  A ) ) ) )
15 ine0 9362 . . . . . . . . . . . . . . . . 17  |-  _i  =/=  0
16 divrec2 9588 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  A
)  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
178, 15, 16mp3an23 1270 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
1810, 17syl 15 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( ( 1  /  _i )  x.  ( _i  x.  A
) ) )
19 irec 11367 . . . . . . . . . . . . . . . 16  |-  ( 1  /  _i )  = 
-u _i
2019oveq1i 5991 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  _i )  x.  ( _i  x.  A ) )  =  ( -u _i  x.  ( _i  x.  A
) )
2118, 20syl6eq 2414 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  ( -u _i  x.  ( _i  x.  A
) ) )
22 divcan3 9595 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  _i  e.  CC  /\  _i  =/=  0 )  ->  (
( _i  x.  A
)  /  _i )  =  A )
238, 15, 22mp3an23 1270 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  /  _i )  =  A )
2421, 23eqtr3d 2400 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2524fveq2d 5636 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
Re `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( Re `  A
) )
2614, 25eqtrd 2398 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
Im `  ( _i  x.  A ) )  =  ( Re `  A
) )
2726eqeq1d 2374 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( Im `  (
_i  x.  A )
)  =  0  <->  (
Re `  A )  =  0 ) )
2812, 27bitrd 244 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Re `  A )  =  0 ) )
2928biimpar 471 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =  0 )  -> 
( _i  x.  A
)  e.  RR )
3029adantlr 695 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  e.  RR )
31 mulne0 9557 . . . . . . . . 9  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( _i  x.  A
)  =/=  0 )
328, 15, 31mpanl12 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  A
)  =/=  0 )
3332adantr 451 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( _i  x.  A )  =/=  0
)
34 rpneg 10534 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  RR  /\  ( _i  x.  A
)  =/=  0 )  ->  ( ( _i  x.  A )  e.  RR+ 
<->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3530, 33, 34syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( (
_i  x.  A )  e.  RR+  <->  -.  -u ( _i  x.  A )  e.  RR+ ) )
3635con2bid 319 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  -.  ( _i  x.  A )  e.  RR+ ) )
37 df-nel 2532 . . . . 5  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
3836, 37syl6bbr 254 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  ( -u (
_i  x.  A )  e.  RR+  <->  ( _i  x.  A )  e/  RR+ )
)
393, 2syl5breqr 4161 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =  0 )  ->  0  <_  ( Re `  A ) )
4039biantrurd 494 . . . 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 271 . . 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 451 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =  0 ) )
4342necon3bbid 2563 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( _i  x.  A )  e.  RR  <->  ( Re `  A )  =/=  0
) )
4443biimpar 471 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR )
45 rpre 10511 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  RR+  ->  ( _i  x.  A )  e.  RR )
4644, 45nsyl 113 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  (
_i  x.  A )  e.  RR+ )
4746, 37sylibr 203 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( _i  x.  A )  e/  RR+ )
4847biantrud 493 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ ) ) )
49 simpr 447 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( Re `  A )  =/=  0
)
5049biantrud 493 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
51 0re 8985 . . . . . . . 8  |-  0  e.  RR
52 recl 11802 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
53 ltlen 9069 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  ( 0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 ) ) )
54 ltnle 9049 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( 0  <  (
Re `  A )  <->  -.  ( Re `  A
)  <_  0 ) )
5553, 54bitr3d 246 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( Re `  A )  e.  RR )  -> 
( ( 0  <_ 
( Re `  A
)  /\  ( Re `  A )  =/=  0
)  <->  -.  ( Re `  A )  <_  0
) )
5651, 52, 55sylancr 644 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  A )  /\  ( Re `  A
)  =/=  0 )  <->  -.  ( Re `  A
)  <_  0 ) )
5756ad2antrr 706 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
Re `  A )  =/=  0 )  <->  -.  (
Re `  A )  <_  0 ) )
5850, 57bitrd 244 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( 0  <_  ( Re `  A )  <->  -.  (
Re `  A )  <_  0 ) )
5948, 58bitr3d 246 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
0  <_  ( Re `  A )  /\  (
_i  x.  A )  e/  RR+ )  <->  -.  (
Re `  A )  <_  0 ) )
60 renegcl 9257 . . . . . . . . . 10  |-  ( -u ( _i  x.  A
)  e.  RR  ->  -u -u ( _i  x.  A
)  e.  RR )
6110negnegd 9295 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  -u -u (
_i  x.  A )  =  ( _i  x.  A ) )
6261eleq1d 2432 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( -u -u ( _i  x.  A
)  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6362ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u -u (
_i  x.  A )  e.  RR  <->  ( _i  x.  A )  e.  RR ) )
6460, 63syl5ib 210 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -u (
_i  x.  A )  e.  RR  ->  ( _i  x.  A )  e.  RR ) )
6544, 64mtod 168 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR )
66 rpre 10511 . . . . . . . 8  |-  ( -u ( _i  x.  A
)  e.  RR+  ->  -u ( _i  x.  A
)  e.  RR )
6765, 66nsyl 113 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -.  -u (
_i  x.  A )  e.  RR+ )
6867, 1sylibr 203 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  -u ( _i  x.  A )  e/  RR+ )
6968biantrud 493 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( (
Re `  A )  <_  0  <->  ( ( Re
`  A )  <_ 
0  /\  -u ( _i  x.  A )  e/  RR+ ) ) )
7069notbid 285 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( Re `  A )  =/=  0
)  ->  ( -.  ( Re `  A )  <_  0  <->  -.  (
( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
7159, 70bitrd 244 . . 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 2607 . 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 11817 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  -u A )  =  -u ( Re `  A ) )
7473breq2d 4137 . . . . . 6  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  0  <_  -u ( Re `  A ) ) )
7552le0neg1d 9491 . . . . . 6  |-  ( A  e.  CC  ->  (
( Re `  A
)  <_  0  <->  0  <_  -u ( Re `  A ) ) )
7674, 75bitr4d 247 . . . . 5  |-  ( A  e.  CC  ->  (
0  <_  ( Re `  -u A )  <->  ( Re `  A )  <_  0
) )
77 mulneg2 9364 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
788, 77mpan 651 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
79 neleq1 2622 . . . . . 6  |-  ( ( _i  x.  -u A
)  =  -u (
_i  x.  A )  ->  ( ( _i  x.  -u A )  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8078, 79syl 15 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  e/  RR+  <->  -u ( _i  x.  A )  e/  RR+ ) )
8176, 80anbi12d 691 . . . 4  |-  ( A  e.  CC  ->  (
( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  ( ( Re `  A
)  <_  0  /\  -u ( _i  x.  A
)  e/  RR+ ) ) )
8281notbid 285 . . 3  |-  ( A  e.  CC  ->  ( -.  ( 0  <_  (
Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )  <->  -.  ( ( Re `  A )  <_  0  /\  -u ( _i  x.  A )  e/  RR+ )
) )
8382adantr 451 . 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 247 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529    e/ wnel 2530   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885   _ici 8886    x. cmul 8889    < clt 9014    <_ cle 9015   -ucneg 9185    / cdiv 9570   RR+crp 10505   Recre 11789   Imcim 11790
This theorem is referenced by:  sqrmo  11944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951  df-rp 10506  df-cj 11791  df-re 11792  df-im 11793
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