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Theorem xov1plusxeqvd 11667
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 11257 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
3 1rp 11225 . . . . . 6  |-  1  e.  RR+
43a1i 11 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR+ )
54, 1rpaddcld 11272 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR+ )
62, 5rerpdivcld 11284 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
75rprecred 11268 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  e.  RR )
8 1red 9612 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR )
9 0red 9598 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  e.  RR )
108, 2readdcld 9624 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR )
118, 1ltaddrpd 11286 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  <  ( 1  +  X ) )
12 recgt1i 10443 . . . . . . . 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 10647 . . . . . 6  |-  ( 1  -  0 )  =  1
1614, 15syl6breqr 4487 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
177, 8, 9, 16ltsub13d 10159 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  -  ( 1  /  ( 1  +  X ) ) ) )
18 ax-1cn 9551 . . . . . . . . 9  |-  1  e.  CC
1918a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
20 xov1plusxeqvd.1 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
2119, 20addcld 9616 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  e.  CC )
2219negcld 9918 . . . . . . . . 9  |-  ( ph  -> 
-u 1  e.  CC )
23 xov1plusxeqvd.2 . . . . . . . . 9  |-  ( ph  ->  X  =/=  -u 1
)
2419, 20, 22, 23addneintrd 9787 . . . . . . . 8  |-  ( ph  ->  ( 1  +  X
)  =/=  ( 1  +  -u 1 ) )
25 1pneg1e0 10645 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
2625a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
2724, 26neeqtrd 2762 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  =/=  0 )
2821, 19, 21, 27divsubdird 10360 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( 1  / 
( 1  +  X
) ) ) )
2919, 20pncan2d 9933 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  -  1 )  =  X )
3029oveq1d 6300 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( X  /  ( 1  +  X ) ) )
3121, 27dividd 10319 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  /  (
1  +  X ) )  =  1 )
3231oveq1d 6300 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  (
1  /  ( 1  +  X ) ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3328, 30, 323eqtr3d 2516 . . . . 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 4473 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( X  /  ( 1  +  X ) ) )
36 1m1e0 10605 . . . . . 6  |-  ( 1  -  1 )  =  0
3713simpld 459 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  /  ( 1  +  X ) ) )
3836, 37syl5eqbr 4480 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  1 )  < 
( 1  /  (
1  +  X ) ) )
398, 8, 7, 38ltsub23d 10158 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  ( 1  / 
( 1  +  X
) ) )  <  1 )
4034, 39eqbrtrd 4467 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  <  1 )
41 0xr 9641 . . . 4  |-  0  e.  RR*
42 1re 9596 . . . . 5  |-  1  e.  RR
4342rexri 9647 . . . 4  |-  1  e.  RR*
44 elioo2 11571 . . . 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 1180 . 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 10318 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  =  ( 1  +  X ) )
5121, 20, 21, 27divsubdird 10360 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( X  / 
( 1  +  X
) ) ) )
5219, 20pncand 9932 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  +  X )  -  X
)  =  1 )
5352oveq1d 6300 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( 1  /  ( 1  +  X ) ) )
5431oveq1d 6300 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  ( X  /  ( 1  +  X ) ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
5551, 53, 543eqtr3d 2516 . . . . . . . . . 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 9612 . . . . . . . . . 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 1008 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
6157, 60resubcld 9988 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  e.  RR )
6256, 61eqeltrd 2555 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR )
63 0red 9598 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  e.  RR )
6459simp3d 1010 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  <  1 )
6564, 15syl6breqr 4487 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
6660, 57, 63, 65ltsub13d 10159 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  -  ( X  /  ( 1  +  X ) ) ) )
6766, 56breqtrrd 4473 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  /  (
1  +  X ) ) )
6862, 67elrpd 11255 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR+ )
6968rprecred 11268 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  e.  RR )
7050, 69eqeltrrd 2556 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  RR )
7170, 57resubcld 9988 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  e.  RR )
7247, 71eqeltrrd 2556 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR )
73 1p0e1 10649 . . . . 5  |-  ( 1  +  0 )  =  1
7459simp2d 1009 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( X  /  (
1  +  X ) ) )
7536, 74syl5eqbr 4480 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  1 )  <  ( X  / 
( 1  +  X
) ) )
7657, 57, 60, 75ltsub23d 10158 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  <  1 )
7756, 76eqbrtrd 4467 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  <  1 )
7868reclt1d 11270 . . . . . . 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 4471 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  +  X
) )
8173, 80syl5eqbr 4480 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  0 )  <  ( 1  +  X ) )
8263, 72, 57ltadd2d 9738 . . . 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 11255 . 2  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR+ )
8546, 84impbida 830 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496   RR*cxr 9628    < clt 9629    - cmin 9806   -ucneg 9807    / cdiv 10207   RR+crp 11221   (,)cioo 11530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-rp 11222  df-ioo 11534
This theorem is referenced by:  angpieqvdlem  22984
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