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Theorem xov1plusxeqvd 11046
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 449 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
21rpred 10653 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
3 1rp 10621 . . . . . 6  |-  1  e.  RR+
43a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR+ )
54, 1rpaddcld 10668 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR+ )
62, 5rerpdivcld 10680 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
75rprecred 10664 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  e.  RR )
8 1re 9095 . . . . . 6  |-  1  e.  RR
98a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR )
10 0re 9096 . . . . . 6  |-  0  e.  RR
1110a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  e.  RR )
129, 2readdcld 9120 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR )
139, 1ltaddrpd 10682 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  <  ( 1  +  X ) )
14 recgt1i 9912 . . . . . . . 8  |-  ( ( ( 1  +  X
)  e.  RR  /\  1  <  ( 1  +  X ) )  -> 
( 0  <  (
1  /  ( 1  +  X ) )  /\  ( 1  / 
( 1  +  X
) )  <  1
) )
1512, 13, 14syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 0  <  ( 1  / 
( 1  +  X
) )  /\  (
1  /  ( 1  +  X ) )  <  1 ) )
1615simprd 451 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  <  1 )
17 ax-1cn 9053 . . . . . . 7  |-  1  e.  CC
1817subid1i 9377 . . . . . 6  |-  ( 1  -  0 )  =  1
1916, 18syl6breqr 4255 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
207, 9, 11, 19ltsub13d 9637 . . . 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 9112 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  e.  CC )
2421negcld 9403 . . . . . . . . 9  |-  ( ph  -> 
-u 1  e.  CC )
25 xov1plusxeqvd.2 . . . . . . . . 9  |-  ( ph  ->  X  =/=  -u 1
)
2621, 22, 24, 25addneintrd 9278 . . . . . . . 8  |-  ( ph  ->  ( 1  +  X
)  =/=  ( 1  +  -u 1 ) )
2717negidi 9374 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
2926, 28neeqtrd 2625 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  =/=  0 )
3023, 21, 23, 29divsubdird 9834 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( 1  / 
( 1  +  X
) ) ) )
3121, 22pncan2d 9418 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  -  1 )  =  X )
3231oveq1d 6099 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( X  /  ( 1  +  X ) ) )
3323, 29dividd 9793 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  /  (
1  +  X ) )  =  1 )
3433oveq1d 6099 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  (
1  /  ( 1  +  X ) ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3530, 32, 343eqtr3d 2478 . . . . 5  |-  ( ph  ->  ( X  /  (
1  +  X ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3635adantr 453 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  =  ( 1  -  (
1  /  ( 1  +  X ) ) ) )
3720, 36breqtrrd 4241 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( X  /  ( 1  +  X ) ) )
38 1m1e0 10073 . . . . . 6  |-  ( 1  -  1 )  =  0
3915simpld 447 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  /  ( 1  +  X ) ) )
4038, 39syl5eqbr 4248 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  1 )  < 
( 1  /  (
1  +  X ) ) )
419, 9, 7, 40ltsub23d 9636 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  ( 1  / 
( 1  +  X
) ) )  <  1 )
4236, 41eqbrtrd 4235 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  <  1 )
43 0xr 9136 . . . 4  |-  0  e.  RR*
448rexri 9142 . . . 4  |-  1  e.  RR*
45 elioo2 10962 . . . 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 655 . . 3  |-  ( ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1 )  <->  ( ( X  /  ( 1  +  X ) )  e.  RR  /\  0  < 
( X  /  (
1  +  X ) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
476, 37, 42, 46syl3anbrc 1139 . 2  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
4831adantr 453 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  =  X )
4923adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  CC )
5029adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  =/=  0 )
5149, 50recrecd 9792 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  =  ( 1  +  X ) )
5223, 22, 23, 29divsubdird 9834 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( X  / 
( 1  +  X
) ) ) )
5321, 22pncand 9417 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  +  X )  -  X
)  =  1 )
5453oveq1d 6099 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( 1  /  ( 1  +  X ) ) )
5533oveq1d 6099 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  ( X  /  ( 1  +  X ) ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
5652, 54, 553eqtr3d 2478 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  (
1  +  X ) )  =  ( 1  -  ( X  / 
( 1  +  X
) ) ) )
5756adantr 453 . . . . . . . . 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 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
6059, 46sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( X  /  (
1  +  X ) )  e.  RR  /\  0  <  ( X  / 
( 1  +  X
) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
6160simp1d 970 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
6258, 61resubcld 9470 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  e.  RR )
6357, 62eqeltrd 2512 . . . . . . . 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 972 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  <  1 )
6665, 18syl6breqr 4255 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
6761, 58, 64, 66ltsub13d 9637 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  -  ( X  /  ( 1  +  X ) ) ) )
6867, 57breqtrrd 4241 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  /  (
1  +  X ) ) )
6963, 68elrpd 10651 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR+ )
7069rprecred 10664 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  e.  RR )
7151, 70eqeltrrd 2513 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  RR )
7271, 58resubcld 9470 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  e.  RR )
7348, 72eqeltrrd 2513 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR )
7417addid1i 9258 . . . . 5  |-  ( 1  +  0 )  =  1
7560simp2d 971 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( X  /  (
1  +  X ) ) )
7638, 75syl5eqbr 4248 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  1 )  <  ( X  / 
( 1  +  X
) ) )
7758, 58, 61, 76ltsub23d 9636 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  <  1 )
7857, 77eqbrtrd 4235 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  <  1 )
7969reclt1d 10666 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  /  (
1  +  X ) )  <  1  <->  1  <  ( 1  / 
( 1  /  (
1  +  X ) ) ) ) )
8078, 79mpbid 203 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  /  (
1  /  ( 1  +  X ) ) ) )
8180, 51breqtrd 4239 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  +  X
) )
8274, 81syl5eqbr 4248 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  0 )  <  ( 1  +  X ) )
8364, 73, 58ltadd2d 9231 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
0  <  X  <->  ( 1  +  0 )  < 
( 1  +  X
) ) )
8482, 83mpbird 225 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  X )
8573, 84elrpd 10651 . 2  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR+ )
8647, 85impbida 807 1  |-  ( ph  ->  ( X  e.  RR+  <->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998   RR*cxr 9124    < clt 9125    - cmin 9296   -ucneg 9297    / cdiv 9682   RR+crp 10617   (,)cioo 10921
This theorem is referenced by:  angpieqvdlem  20674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-rp 10618  df-ioo 10925
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