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Theorem xov1plusxeqvd 11431
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 461 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
21rpred 11027 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
3 1rp 10995 . . . . . 6  |-  1  e.  RR+
43a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR+ )
54, 1rpaddcld 11042 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR+ )
62, 5rerpdivcld 11054 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
75rprecred 11038 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  e.  RR )
8 1red 9401 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR )
9 0red 9387 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  e.  RR )
108, 2readdcld 9413 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR )
118, 1ltaddrpd 11056 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  <  ( 1  +  X ) )
12 recgt1i 10229 . . . . . . . 8  |-  ( ( ( 1  +  X
)  e.  RR  /\  1  <  ( 1  +  X ) )  -> 
( 0  <  (
1  /  ( 1  +  X ) )  /\  ( 1  / 
( 1  +  X
) )  <  1
) )
1310, 11, 12syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 0  <  ( 1  / 
( 1  +  X
) )  /\  (
1  /  ( 1  +  X ) )  <  1 ) )
1413simprd 463 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  <  1 )
15 1m0e1 10432 . . . . . 6  |-  ( 1  -  0 )  =  1
1614, 15syl6breqr 4332 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
177, 8, 9, 16ltsub13d 9945 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  -  ( 1  /  ( 1  +  X ) ) ) )
18 ax-1cn 9340 . . . . . . . . 9  |-  1  e.  CC
1918a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
20 xov1plusxeqvd.1 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
2119, 20addcld 9405 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  e.  CC )
2219negcld 9706 . . . . . . . . 9  |-  ( ph  -> 
-u 1  e.  CC )
23 xov1plusxeqvd.2 . . . . . . . . 9  |-  ( ph  ->  X  =/=  -u 1
)
2419, 20, 22, 23addneintrd 9576 . . . . . . . 8  |-  ( ph  ->  ( 1  +  X
)  =/=  ( 1  +  -u 1 ) )
25 1pneg1e0 10430 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
2625a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
2724, 26neeqtrd 2630 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  =/=  0 )
2821, 19, 21, 27divsubdird 10146 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( 1  / 
( 1  +  X
) ) ) )
2919, 20pncan2d 9721 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  -  1 )  =  X )
3029oveq1d 6106 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( X  /  ( 1  +  X ) ) )
3121, 27dividd 10105 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  /  (
1  +  X ) )  =  1 )
3231oveq1d 6106 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  (
1  /  ( 1  +  X ) ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3328, 30, 323eqtr3d 2483 . . . . 5  |-  ( ph  ->  ( X  /  (
1  +  X ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3433adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  =  ( 1  -  (
1  /  ( 1  +  X ) ) ) )
3517, 34breqtrrd 4318 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( X  /  ( 1  +  X ) ) )
36 1m1e0 10390 . . . . . 6  |-  ( 1  -  1 )  =  0
3713simpld 459 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  /  ( 1  +  X ) ) )
3836, 37syl5eqbr 4325 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  1 )  < 
( 1  /  (
1  +  X ) ) )
398, 8, 7, 38ltsub23d 9944 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  ( 1  / 
( 1  +  X
) ) )  <  1 )
4034, 39eqbrtrd 4312 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  <  1 )
41 0xr 9430 . . . 4  |-  0  e.  RR*
42 1re 9385 . . . . 5  |-  1  e.  RR
4342rexri 9436 . . . 4  |-  1  e.  RR*
44 elioo2 11341 . . . 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 ) ) )
4541, 43, 44mp2an 672 . . 3  |-  ( ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1 )  <->  ( ( X  /  ( 1  +  X ) )  e.  RR  /\  0  < 
( X  /  (
1  +  X ) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
466, 35, 40, 45syl3anbrc 1172 . 2  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
4729adantr 465 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  =  X )
4821adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  CC )
4927adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  =/=  0 )
5048, 49recrecd 10104 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  =  ( 1  +  X ) )
5121, 20, 21, 27divsubdird 10146 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( X  / 
( 1  +  X
) ) ) )
5219, 20pncand 9720 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  +  X )  -  X
)  =  1 )
5352oveq1d 6106 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( 1  /  ( 1  +  X ) ) )
5431oveq1d 6106 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  ( X  /  ( 1  +  X ) ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
5551, 53, 543eqtr3d 2483 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  (
1  +  X ) )  =  ( 1  -  ( X  / 
( 1  +  X
) ) ) )
5655adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
57 1red 9401 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
58 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
5958, 45sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( X  /  (
1  +  X ) )  e.  RR  /\  0  <  ( X  / 
( 1  +  X
) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
6059simp1d 1000 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
6157, 60resubcld 9776 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  e.  RR )
6256, 61eqeltrd 2517 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR )
63 0red 9387 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  e.  RR )
6459simp3d 1002 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  <  1 )
6564, 15syl6breqr 4332 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
6660, 57, 63, 65ltsub13d 9945 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  -  ( X  /  ( 1  +  X ) ) ) )
6766, 56breqtrrd 4318 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  /  (
1  +  X ) ) )
6862, 67elrpd 11025 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR+ )
6968rprecred 11038 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  e.  RR )
7050, 69eqeltrrd 2518 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  RR )
7170, 57resubcld 9776 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  e.  RR )
7247, 71eqeltrrd 2518 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR )
73 1p0e1 10434 . . . . 5  |-  ( 1  +  0 )  =  1
7459simp2d 1001 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( X  /  (
1  +  X ) ) )
7536, 74syl5eqbr 4325 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  1 )  <  ( X  / 
( 1  +  X
) ) )
7657, 57, 60, 75ltsub23d 9944 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  <  1 )
7756, 76eqbrtrd 4312 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  <  1 )
7868reclt1d 11040 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  /  (
1  +  X ) )  <  1  <->  1  <  ( 1  / 
( 1  /  (
1  +  X ) ) ) ) )
7977, 78mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  /  (
1  /  ( 1  +  X ) ) ) )
8079, 50breqtrd 4316 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  +  X
) )
8173, 80syl5eqbr 4325 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  0 )  <  ( 1  +  X ) )
8263, 72, 57ltadd2d 9527 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
0  <  X  <->  ( 1  +  0 )  < 
( 1  +  X
) ) )
8381, 82mpbird 232 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  X )
8472, 83elrpd 11025 . 2  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR+ )
8546, 84impbida 828 1  |-  ( ph  ->  ( X  e.  RR+  <->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    - cmin 9595   -ucneg 9596    / cdiv 9993   RR+crp 10991   (,)cioo 11300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-rp 10992  df-ioo 11304
This theorem is referenced by:  angpieqvdlem  22223
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