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Theorem xov1plusxeqvd 11033
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
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
21rpred 10640 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
3 1rp 10608 . . . . . 6  |-  1  e.  RR+
43a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR+ )
54, 1rpaddcld 10655 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR+ )
62, 5rerpdivcld 10667 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
75rprecred 10651 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  e.  RR )
8 1re 9082 . . . . . 6  |-  1  e.  RR
98a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR )
10 0re 9083 . . . . . 6  |-  0  e.  RR
1110a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  e.  RR )
129, 2readdcld 9107 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR )
139, 1ltaddrpd 10669 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  <  ( 1  +  X ) )
14 recgt1i 9899 . . . . . . . 8  |-  ( ( ( 1  +  X
)  e.  RR  /\  1  <  ( 1  +  X ) )  -> 
( 0  <  (
1  /  ( 1  +  X ) )  /\  ( 1  / 
( 1  +  X
) )  <  1
) )
1512, 13, 14syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 0  <  ( 1  / 
( 1  +  X
) )  /\  (
1  /  ( 1  +  X ) )  <  1 ) )
1615simprd 450 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  <  1 )
17 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
1817subid1i 9364 . . . . . 6  |-  ( 1  -  0 )  =  1
1916, 18syl6breqr 4244 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
207, 9, 11, 19ltsub13d 9624 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  -  ( 1  /  ( 1  +  X ) ) ) )
2117a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
22 xov1plusxeqvd.1 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
2321, 22addcld 9099 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  e.  CC )
2421negcld 9390 . . . . . . . . 9  |-  ( ph  -> 
-u 1  e.  CC )
25 xov1plusxeqvd.2 . . . . . . . . 9  |-  ( ph  ->  X  =/=  -u 1
)
2621, 22, 24, 25addneintrd 9265 . . . . . . . 8  |-  ( ph  ->  ( 1  +  X
)  =/=  ( 1  +  -u 1 ) )
2717negidi 9361 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
2926, 28neeqtrd 2620 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  =/=  0 )
3023, 21, 23, 29divsubdird 9821 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( 1  / 
( 1  +  X
) ) ) )
3121, 22pncan2d 9405 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  -  1 )  =  X )
3231oveq1d 6088 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( X  /  ( 1  +  X ) ) )
3323, 29dividd 9780 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  /  (
1  +  X ) )  =  1 )
3433oveq1d 6088 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  (
1  /  ( 1  +  X ) ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3530, 32, 343eqtr3d 2475 . . . . 5  |-  ( ph  ->  ( X  /  (
1  +  X ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3635adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  =  ( 1  -  (
1  /  ( 1  +  X ) ) ) )
3720, 36breqtrrd 4230 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( X  /  ( 1  +  X ) ) )
38 1m1e0 10060 . . . . . 6  |-  ( 1  -  1 )  =  0
3915simpld 446 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  /  ( 1  +  X ) ) )
4038, 39syl5eqbr 4237 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  1 )  < 
( 1  /  (
1  +  X ) ) )
419, 9, 7, 40ltsub23d 9623 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  ( 1  / 
( 1  +  X
) ) )  <  1 )
4236, 41eqbrtrd 4224 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  <  1 )
43 0xr 9123 . . . 4  |-  0  e.  RR*
448rexri 9129 . . . 4  |-  1  e.  RR*
45 elioo2 10949 . . . 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 ) ) )
4643, 44, 45mp2an 654 . . 3  |-  ( ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1 )  <->  ( ( X  /  ( 1  +  X ) )  e.  RR  /\  0  < 
( X  /  (
1  +  X ) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
476, 37, 42, 46syl3anbrc 1138 . 2  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
4831adantr 452 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  =  X )
4923adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  CC )
5029adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  =/=  0 )
5149, 50recrecd 9779 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  =  ( 1  +  X ) )
5223, 22, 23, 29divsubdird 9821 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( X  / 
( 1  +  X
) ) ) )
5321, 22pncand 9404 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  +  X )  -  X
)  =  1 )
5453oveq1d 6088 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( 1  /  ( 1  +  X ) ) )
5533oveq1d 6088 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  ( X  /  ( 1  +  X ) ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
5652, 54, 553eqtr3d 2475 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  (
1  +  X ) )  =  ( 1  -  ( X  / 
( 1  +  X
) ) ) )
5756adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
588a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
59 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
6059, 46sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( X  /  (
1  +  X ) )  e.  RR  /\  0  <  ( X  / 
( 1  +  X
) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
6160simp1d 969 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
6258, 61resubcld 9457 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  e.  RR )
6357, 62eqeltrd 2509 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR )
6410a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  e.  RR )
6560simp3d 971 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  <  1 )
6665, 18syl6breqr 4244 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
6761, 58, 64, 66ltsub13d 9624 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  -  ( X  /  ( 1  +  X ) ) ) )
6867, 57breqtrrd 4230 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  /  (
1  +  X ) ) )
6963, 68elrpd 10638 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR+ )
7069rprecred 10651 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  e.  RR )
7151, 70eqeltrrd 2510 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  RR )
7271, 58resubcld 9457 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  e.  RR )
7348, 72eqeltrrd 2510 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR )
7417addid1i 9245 . . . . 5  |-  ( 1  +  0 )  =  1
7560simp2d 970 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( X  /  (
1  +  X ) ) )
7638, 75syl5eqbr 4237 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  1 )  <  ( X  / 
( 1  +  X
) ) )
7758, 58, 61, 76ltsub23d 9623 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  <  1 )
7857, 77eqbrtrd 4224 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  <  1 )
7969reclt1d 10653 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  /  (
1  +  X ) )  <  1  <->  1  <  ( 1  / 
( 1  /  (
1  +  X ) ) ) ) )
8078, 79mpbid 202 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  /  (
1  /  ( 1  +  X ) ) ) )
8180, 51breqtrd 4228 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  +  X
) )
8274, 81syl5eqbr 4237 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  0 )  <  ( 1  +  X ) )
8364, 73, 58ltadd2d 9218 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
0  <  X  <->  ( 1  +  0 )  < 
( 1  +  X
) ) )
8482, 83mpbird 224 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  X )
8573, 84elrpd 10638 . 2  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR+ )
8647, 85impbida 806 1  |-  ( ph  ->  ( X  e.  RR+  <->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985   RR*cxr 9111    < clt 9112    - cmin 9283   -ucneg 9284    / cdiv 9669   RR+crp 10604   (,)cioo 10908
This theorem is referenced by:  angpieqvdlem  20661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-rp 10605  df-ioo 10912
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