Theorem cnpart 13980

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

Step	Hyp	Ref	Expression
1		df-nel 2898	. . . . . 6 |- ( -u ( _i x. A ) e/ RR+ <-> -. -u ( _i x. A ) e. RR+ )
2		simpr 477	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( Re ` A ) = 0 )
3		0le0 11110	. . . . . . . 8 |- 0 <_ 0
4	2, 3	syl6eqbr 4692	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( Re ` A ) <_ 0 )
5	4	biantrurd 529	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( -u ( _i x. A ) e/ RR+ <-> ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
6	1, 5	syl5bbr 274	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( -. -u ( _i x. A ) e. RR+ <-> ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
7	6	con1bid 345	. . . 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 9995	. . . . . . . . . . . 12 |- _i e. CC
9		mulcl 10020	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
10	8, 9	mpan 706	. . . . . . . . . . 11 |- ( A e. CC -> ( _i x. A ) e. CC )
11		reim0b 13859	. . . . . . . . . . 11 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) e. RR <-> ( Im ` ( _i x. A ) ) = 0 ) )
12	10, 11	syl 17	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) e. RR <-> ( Im ` ( _i x. A ) ) = 0 ) )
13		imre 13848	. . . . . . . . . . . . 13 |- ( ( _i x. A ) e. CC -> ( Im ` ( _i x. A ) ) = ( Re ` ( -u _i x. ( _i x. A ) ) ) )
14	10, 13	syl 17	. . . . . . . . . . . 12 |- ( A e. CC -> ( Im ` ( _i x. A ) ) = ( Re ` ( -u _i x. ( _i x. A ) ) ) )
15		ine0 10465	. . . . . . . . . . . . . . . . 17 |- _i =/= 0
16		divrec2 10702	. . . . . . . . . . . . . . . . 17 |- ( ( ( _i x. A ) e. CC /\ _i e. CC /\ _i =/= 0 ) -> ( ( _i x. A ) / _i ) = ( ( 1 / _i ) x. ( _i x. A ) ) )
17	8, 15, 16	mp3an23 1416	. . . . . . . . . . . . . . . 16 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) / _i ) = ( ( 1 / _i ) x. ( _i x. A ) ) )
18	10, 17	syl 17	. . . . . . . . . . . . . . 15 |- ( A e. CC -> ( ( _i x. A ) / _i ) = ( ( 1 / _i ) x. ( _i x. A ) ) )
19		irec 12964	. . . . . . . . . . . . . . . 16 |- ( 1 / _i ) = -u _i
20	19	oveq1i 6660	. . . . . . . . . . . . . . 15 |- ( ( 1 / _i ) x. ( _i x. A ) ) = ( -u _i x. ( _i x. A ) )
21	18, 20	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( A e. CC -> ( ( _i x. A ) / _i ) = ( -u _i x. ( _i x. A ) ) )
22		divcan3 10711	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ _i e. CC /\ _i =/= 0 ) -> ( ( _i x. A ) / _i ) = A )
23	8, 15, 22	mp3an23 1416	. . . . . . . . . . . . . 14 |- ( A e. CC -> ( ( _i x. A ) / _i ) = A )
24	21, 23	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( A e. CC -> ( -u _i x. ( _i x. A ) ) = A )
25	24	fveq2d 6195	. . . . . . . . . . . 12 |- ( A e. CC -> ( Re ` ( -u _i x. ( _i x. A ) ) ) = ( Re ` A ) )
26	14, 25	eqtrd 2656	. . . . . . . . . . 11 |- ( A e. CC -> ( Im ` ( _i x. A ) ) = ( Re ` A ) )
27	26	eqeq1d 2624	. . . . . . . . . 10 |- ( A e. CC -> ( ( Im ` ( _i x. A ) ) = 0 <-> ( Re ` A ) = 0 ) )
28	12, 27	bitrd 268	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) e. RR <-> ( Re ` A ) = 0 ) )
29	28	biimpar 502	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) = 0 ) -> ( _i x. A ) e. RR )
30	29	adantlr 751	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( _i x. A ) e. RR )
31		mulne0 10669	. . . . . . . . 9 |- ( ( ( _i e. CC /\ _i =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( _i x. A ) =/= 0 )
32	8, 15, 31	mpanl12 718	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( _i x. A ) =/= 0 )
33	32	adantr 481	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( _i x. A ) =/= 0 )
34		rpneg 11863	. . . . . . 7 |- ( ( ( _i x. A ) e. RR /\ ( _i x. A ) =/= 0 ) -> ( ( _i x. A ) e. RR+ <-> -. -u ( _i x. A ) e. RR+ ) )
35	30, 33, 34	syl2anc 693	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( ( _i x. A ) e. RR+ <-> -. -u ( _i x. A ) e. RR+ ) )
36	35	con2bid 344	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( -u ( _i x. A ) e. RR+ <-> -. ( _i x. A ) e. RR+ ) )
37		df-nel 2898	. . . . 5 |- ( ( _i x. A ) e/ RR+ <-> -. ( _i x. A ) e. RR+ )
38	36, 37	syl6bbr 278	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( -u ( _i x. A ) e. RR+ <-> ( _i x. A ) e/ RR+ ) )
39	3, 2	syl5breqr 4691	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> 0 <_ ( Re ` A ) )
40	39	biantrurd 529	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) = 0 ) -> ( ( _i x. A ) e/ RR+ <-> ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) ) )
41	7, 38, 40	3bitrrd 295	. . 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+ ) ) )
42	28	adantr 481	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( _i x. A ) e. RR <-> ( Re ` A ) = 0 ) )
43	42	necon3bbid 2831	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( _i x. A ) e. RR <-> ( Re ` A ) =/= 0 ) )
44	43	biimpar 502	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> -. ( _i x. A ) e. RR )
45		rpre 11839	. . . . . . . 8 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
46	44, 45	nsyl 135	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> -. ( _i x. A ) e. RR+ )
47	46, 37	sylibr 224	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( _i x. A ) e/ RR+ )
48	47	biantrud 528	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( 0 <_ ( Re ` A ) <-> ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) ) )
49		simpr 477	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( Re ` A ) =/= 0 )
50	49	biantrud 528	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( 0 <_ ( Re ` A ) <-> ( 0 <_ ( Re ` A ) /\ ( Re ` A ) =/= 0 ) ) )
51		0re 10040	. . . . . . . 8 |- 0 e. RR
52		recl 13850	. . . . . . . 8 |- ( A e. CC -> ( Re ` A ) e. RR )
53		ltlen 10138	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( 0 < ( Re ` A ) <-> ( 0 <_ ( Re ` A ) /\ ( Re ` A ) =/= 0 ) ) )
54		ltnle 10117	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( 0 < ( Re ` A ) <-> -. ( Re ` A ) <_ 0 ) )
55	53, 54	bitr3d 270	. . . . . . . 8 |- ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( ( 0 <_ ( Re ` A ) /\ ( Re ` A ) =/= 0 ) <-> -. ( Re ` A ) <_ 0 ) )
56	51, 52, 55	sylancr 695	. . . . . . 7 |- ( A e. CC -> ( ( 0 <_ ( Re ` A ) /\ ( Re ` A ) =/= 0 ) <-> -. ( Re ` A ) <_ 0 ) )
57	56	ad2antrr 762	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( ( 0 <_ ( Re ` A ) /\ ( Re ` A ) =/= 0 ) <-> -. ( Re ` A ) <_ 0 ) )
58	50, 57	bitrd 268	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( 0 <_ ( Re ` A ) <-> -. ( Re ` A ) <_ 0 ) )
59	48, 58	bitr3d 270	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) <-> -. ( Re ` A ) <_ 0 ) )
60		renegcl 10344	. . . . . . . . . 10 |- ( -u ( _i x. A ) e. RR -> -u -u ( _i x. A ) e. RR )
61	10	negnegd 10383	. . . . . . . . . . . 12 |- ( A e. CC -> -u -u ( _i x. A ) = ( _i x. A ) )
62	61	eleq1d 2686	. . . . . . . . . . 11 |- ( A e. CC -> ( -u -u ( _i x. A ) e. RR <-> ( _i x. A ) e. RR ) )
63	62	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( -u -u ( _i x. A ) e. RR <-> ( _i x. A ) e. RR ) )
64	60, 63	syl5ib 234	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( -u ( _i x. A ) e. RR -> ( _i x. A ) e. RR ) )
65	44, 64	mtod 189	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> -. -u ( _i x. A ) e. RR )
66		rpre 11839	. . . . . . . 8 |- ( -u ( _i x. A ) e. RR+ -> -u ( _i x. A ) e. RR )
67	65, 66	nsyl 135	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> -. -u ( _i x. A ) e. RR+ )
68	67, 1	sylibr 224	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> -u ( _i x. A ) e/ RR+ )
69	68	biantrud 528	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( ( Re ` A ) <_ 0 <-> ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
70	69	notbid 308	. . . 4 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( Re ` A ) =/= 0 ) -> ( -. ( Re ` A ) <_ 0 <-> -. ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
71	59, 70	bitrd 268	. . 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+ ) ) )
72	41, 71	pm2.61dane 2881	. 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 13865	. . . . . . 7 |- ( A e. CC -> ( Re ` -u A ) = -u ( Re ` A ) )
74	73	breq2d 4665	. . . . . 6 |- ( A e. CC -> ( 0 <_ ( Re ` -u A ) <-> 0 <_ -u ( Re ` A ) ) )
75	52	le0neg1d 10599	. . . . . 6 |- ( A e. CC -> ( ( Re ` A ) <_ 0 <-> 0 <_ -u ( Re ` A ) ) )
76	74, 75	bitr4d 271	. . . . 5 |- ( A e. CC -> ( 0 <_ ( Re ` -u A ) <-> ( Re ` A ) <_ 0 ) )
77		mulneg2 10467	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) )
78	8, 77	mpan 706	. . . . . 6 |- ( A e. CC -> ( _i x. -u A ) = -u ( _i x. A ) )
79		neleq1 2902	. . . . . 6 |- ( ( _i x. -u A ) = -u ( _i x. A ) -> ( ( _i x. -u A ) e/ RR+ <-> -u ( _i x. A ) e/ RR+ ) )
80	78, 79	syl 17	. . . . 5 |- ( A e. CC -> ( ( _i x. -u A ) e/ RR+ <-> -u ( _i x. A ) e/ RR+ ) )
81	76, 80	anbi12d 747	. . . 4 |- ( A e. CC -> ( ( 0 <_ ( Re ` -u A ) /\ ( _i x. -u A ) e/ RR+ ) <-> ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
82	81	notbid 308	. . 3 |- ( A e. CC -> ( -. ( 0 <_ ( Re ` -u A ) /\ ( _i x. -u A ) e/ RR+ ) <-> -. ( ( Re ` A ) <_ 0 /\ -u ( _i x. A ) e/ RR+ ) ) )
83	82	adantr 481	. 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+ ) ) )
84	72, 83	bitr4d 271	1 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( 0 <_ ( Re ` A ) /\ ( _i x. A ) e/ RR+ ) <-> -. ( 0 <_ ( Re ` -u A ) /\ ( _i x. -u A ) e/ RR+ ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   RR+crp 11832   Recre 13837   Imcim 13838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by:  sqrmo  13992