Theorem xov1plusxeqvd 12318

Description: A complex number X is positive real iff X / ( 1 + X ) is in ( 0 (,) 1 ). Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
xov1plusxeqvd.1	|- ( ph -> X e. CC )
xov1plusxeqvd.2	|- ( ph -> X =/= -u 1 )

Assertion

Ref	Expression
xov1plusxeqvd	|- ( ph -> ( X e. RR+ <-> ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) )

Proof of Theorem xov1plusxeqvd

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> X e. RR+ )
2	1	rpred 11872	. . . 4 |- ( ( ph /\ X e. RR+ ) -> X e. RR )
3		1rp 11836	. . . . . 6 |- 1 e. RR+
4	3	a1i 11	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> 1 e. RR+ )
5	4, 1	rpaddcld 11887	. . . 4 |- ( ( ph /\ X e. RR+ ) -> ( 1 + X ) e. RR+ )
6	2, 5	rerpdivcld 11903	. . 3 |- ( ( ph /\ X e. RR+ ) -> ( X / ( 1 + X ) ) e. RR )
7	5	rprecred 11883	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> ( 1 / ( 1 + X ) ) e. RR )
8		1red 10055	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> 1 e. RR )
9		0red 10041	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> 0 e. RR )
10	8, 2	readdcld 10069	. . . . . . . 8 |- ( ( ph /\ X e. RR+ ) -> ( 1 + X ) e. RR )
11	8, 1	ltaddrpd 11905	. . . . . . . 8 |- ( ( ph /\ X e. RR+ ) -> 1 < ( 1 + X ) )
12		recgt1i 10920	. . . . . . . 8 |- ( ( ( 1 + X ) e. RR /\ 1 < ( 1 + X ) ) -> ( 0 < ( 1 / ( 1 + X ) ) /\ ( 1 / ( 1 + X ) ) < 1 ) )
13	10, 11, 12	syl2anc 693	. . . . . . 7 |- ( ( ph /\ X e. RR+ ) -> ( 0 < ( 1 / ( 1 + X ) ) /\ ( 1 / ( 1 + X ) ) < 1 ) )
14	13	simprd 479	. . . . . 6 |- ( ( ph /\ X e. RR+ ) -> ( 1 / ( 1 + X ) ) < 1 )
15		1m0e1 11131	. . . . . 6 |- ( 1 - 0 ) = 1
16	14, 15	syl6breqr 4695	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> ( 1 / ( 1 + X ) ) < ( 1 - 0 ) )
17	7, 8, 9, 16	ltsub13d 10633	. . . 4 |- ( ( ph /\ X e. RR+ ) -> 0 < ( 1 - ( 1 / ( 1 + X ) ) ) )
18		1cnd 10056	. . . . . . . 8 |- ( ph -> 1 e. CC )
19		xov1plusxeqvd.1	. . . . . . . 8 |- ( ph -> X e. CC )
20	18, 19	addcld 10059	. . . . . . 7 |- ( ph -> ( 1 + X ) e. CC )
21	18	negcld 10379	. . . . . . . . 9 |- ( ph -> -u 1 e. CC )
22		xov1plusxeqvd.2	. . . . . . . . 9 |- ( ph -> X =/= -u 1 )
23	18, 19, 21, 22	addneintrd 10243	. . . . . . . 8 |- ( ph -> ( 1 + X ) =/= ( 1 + -u 1 ) )
24		1pneg1e0 11129	. . . . . . . . 9 |- ( 1 + -u 1 ) = 0
25	24	a1i 11	. . . . . . . 8 |- ( ph -> ( 1 + -u 1 ) = 0 )
26	23, 25	neeqtrd 2863	. . . . . . 7 |- ( ph -> ( 1 + X ) =/= 0 )
27	20, 18, 20, 26	divsubdird 10840	. . . . . 6 |- ( ph -> ( ( ( 1 + X ) - 1 ) / ( 1 + X ) ) = ( ( ( 1 + X ) / ( 1 + X ) ) - ( 1 / ( 1 + X ) ) ) )
28	18, 19	pncan2d 10394	. . . . . . 7 |- ( ph -> ( ( 1 + X ) - 1 ) = X )
29	28	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( 1 + X ) - 1 ) / ( 1 + X ) ) = ( X / ( 1 + X ) ) )
30	20, 26	dividd 10799	. . . . . . 7 |- ( ph -> ( ( 1 + X ) / ( 1 + X ) ) = 1 )
31	30	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( 1 + X ) / ( 1 + X ) ) - ( 1 / ( 1 + X ) ) ) = ( 1 - ( 1 / ( 1 + X ) ) ) )
32	27, 29, 31	3eqtr3d 2664	. . . . 5 |- ( ph -> ( X / ( 1 + X ) ) = ( 1 - ( 1 / ( 1 + X ) ) ) )
33	32	adantr 481	. . . 4 |- ( ( ph /\ X e. RR+ ) -> ( X / ( 1 + X ) ) = ( 1 - ( 1 / ( 1 + X ) ) ) )
34	17, 33	breqtrrd 4681	. . 3 |- ( ( ph /\ X e. RR+ ) -> 0 < ( X / ( 1 + X ) ) )
35		1m1e0 11089	. . . . . 6 |- ( 1 - 1 ) = 0
36	13	simpld 475	. . . . . 6 |- ( ( ph /\ X e. RR+ ) -> 0 < ( 1 / ( 1 + X ) ) )
37	35, 36	syl5eqbr 4688	. . . . 5 |- ( ( ph /\ X e. RR+ ) -> ( 1 - 1 ) < ( 1 / ( 1 + X ) ) )
38	8, 8, 7, 37	ltsub23d 10632	. . . 4 |- ( ( ph /\ X e. RR+ ) -> ( 1 - ( 1 / ( 1 + X ) ) ) < 1 )
39	33, 38	eqbrtrd 4675	. . 3 |- ( ( ph /\ X e. RR+ ) -> ( X / ( 1 + X ) ) < 1 )
40		0xr 10086	. . . 4 |- 0 e. RR*
41		1re 10039	. . . . 5 |- 1 e. RR
42	41	rexri 10097	. . . 4 |- 1 e. RR*
43		elioo2 12216	. . . 4 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) <-> ( ( X / ( 1 + X ) ) e. RR /\ 0 < ( X / ( 1 + X ) ) /\ ( X / ( 1 + X ) ) < 1 ) ) )
44	40, 42, 43	mp2an 708	. . 3 |- ( ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) <-> ( ( X / ( 1 + X ) ) e. RR /\ 0 < ( X / ( 1 + X ) ) /\ ( X / ( 1 + X ) ) < 1 ) )
45	6, 34, 39, 44	syl3anbrc 1246	. 2 |- ( ( ph /\ X e. RR+ ) -> ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) )
46	28	adantr 481	. . . 4 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( ( 1 + X ) - 1 ) = X )
47	20	adantr 481	. . . . . . 7 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 + X ) e. CC )
48	26	adantr 481	. . . . . . 7 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 + X ) =/= 0 )
49	47, 48	recrecd 10798	. . . . . 6 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 / ( 1 + X ) ) ) = ( 1 + X ) )
50	20, 19, 20, 26	divsubdird 10840	. . . . . . . . . . 11 |- ( ph -> ( ( ( 1 + X ) - X ) / ( 1 + X ) ) = ( ( ( 1 + X ) / ( 1 + X ) ) - ( X / ( 1 + X ) ) ) )
51	18, 19	pncand 10393	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 + X ) - X ) = 1 )
52	51	oveq1d 6665	. . . . . . . . . . 11 |- ( ph -> ( ( ( 1 + X ) - X ) / ( 1 + X ) ) = ( 1 / ( 1 + X ) ) )
53	30	oveq1d 6665	. . . . . . . . . . 11 |- ( ph -> ( ( ( 1 + X ) / ( 1 + X ) ) - ( X / ( 1 + X ) ) ) = ( 1 - ( X / ( 1 + X ) ) ) )
54	50, 52, 53	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( 1 / ( 1 + X ) ) = ( 1 - ( X / ( 1 + X ) ) ) )
55	54	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 + X ) ) = ( 1 - ( X / ( 1 + X ) ) ) )
56		1red 10055	. . . . . . . . . 10 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 1 e. RR )
57		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) )
58	57, 44	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( ( X / ( 1 + X ) ) e. RR /\ 0 < ( X / ( 1 + X ) ) /\ ( X / ( 1 + X ) ) < 1 ) )
59	58	simp1d 1073	. . . . . . . . . 10 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( X / ( 1 + X ) ) e. RR )
60	56, 59	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 - ( X / ( 1 + X ) ) ) e. RR )
61	55, 60	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 + X ) ) e. RR )
62		0red 10041	. . . . . . . . . 10 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 0 e. RR )
63	58	simp3d 1075	. . . . . . . . . . 11 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( X / ( 1 + X ) ) < 1 )
64	63, 15	syl6breqr 4695	. . . . . . . . . 10 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( X / ( 1 + X ) ) < ( 1 - 0 ) )
65	59, 56, 62, 64	ltsub13d 10633	. . . . . . . . 9 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 0 < ( 1 - ( X / ( 1 + X ) ) ) )
66	65, 55	breqtrrd 4681	. . . . . . . 8 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 0 < ( 1 / ( 1 + X ) ) )
67	61, 66	elrpd 11869	. . . . . . 7 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 + X ) ) e. RR+ )
68	67	rprecred 11883	. . . . . 6 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 / ( 1 + X ) ) ) e. RR )
69	49, 68	eqeltrrd 2702	. . . . 5 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 + X ) e. RR )
70	69, 56	resubcld 10458	. . . 4 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( ( 1 + X ) - 1 ) e. RR )
71	46, 70	eqeltrrd 2702	. . 3 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> X e. RR )
72		1p0e1 11133	. . . . 5 |- ( 1 + 0 ) = 1
73	58	simp2d 1074	. . . . . . . . . 10 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 0 < ( X / ( 1 + X ) ) )
74	35, 73	syl5eqbr 4688	. . . . . . . . 9 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 - 1 ) < ( X / ( 1 + X ) ) )
75	56, 56, 59, 74	ltsub23d 10632	. . . . . . . 8 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 - ( X / ( 1 + X ) ) ) < 1 )
76	55, 75	eqbrtrd 4675	. . . . . . 7 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 / ( 1 + X ) ) < 1 )
77	67	reclt1d 11885	. . . . . . 7 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( ( 1 / ( 1 + X ) ) < 1 <-> 1 < ( 1 / ( 1 / ( 1 + X ) ) ) ) )
78	76, 77	mpbid 222	. . . . . 6 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 1 < ( 1 / ( 1 / ( 1 + X ) ) ) )
79	78, 49	breqtrd 4679	. . . . 5 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 1 < ( 1 + X ) )
80	72, 79	syl5eqbr 4688	. . . 4 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 1 + 0 ) < ( 1 + X ) )
81	62, 71, 56	ltadd2d 10193	. . . 4 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> ( 0 < X <-> ( 1 + 0 ) < ( 1 + X ) ) )
82	80, 81	mpbird 247	. . 3 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> 0 < X )
83	71, 82	elrpd 11869	. 2 |- ( ( ph /\ ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) -> X e. RR+ )
84	45, 83	impbida 877	1 |- ( ph -> ( X e. RR+ <-> ( X / ( 1 + X ) ) e. ( 0 (,) 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   RR+crp 11832   (,)cioo 12175

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-cnex 9992  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-iun 4522  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-1st 7168  df-2nd 7169  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-rp 11833  df-ioo 12179

This theorem is referenced by:  angpieqvdlem  24555