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Theorem xov1plusxeqvd 11669
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 459 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
21rpred 11259 . . . 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 11274 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR+ )
62, 5rerpdivcld 11286 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
75rprecred 11270 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  e.  RR )
8 1red 9600 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  e.  RR )
9 0red 9586 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  e.  RR )
108, 2readdcld 9612 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  +  X )  e.  RR )
118, 1ltaddrpd 11288 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  1  <  ( 1  +  X ) )
12 recgt1i 10437 . . . . . . . 8  |-  ( ( ( 1  +  X
)  e.  RR  /\  1  <  ( 1  +  X ) )  -> 
( 0  <  (
1  /  ( 1  +  X ) )  /\  ( 1  / 
( 1  +  X
) )  <  1
) )
1310, 11, 12syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 0  <  ( 1  / 
( 1  +  X
) )  /\  (
1  /  ( 1  +  X ) )  <  1 ) )
1413simprd 461 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  <  1 )
15 1m0e1 10642 . . . . . 6  |-  ( 1  -  0 )  =  1
1614, 15syl6breqr 4479 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
177, 8, 9, 16ltsub13d 10154 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  -  ( 1  /  ( 1  +  X ) ) ) )
18 1cnd 9601 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
19 xov1plusxeqvd.1 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
2018, 19addcld 9604 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  e.  CC )
2118negcld 9909 . . . . . . . . 9  |-  ( ph  -> 
-u 1  e.  CC )
22 xov1plusxeqvd.2 . . . . . . . . 9  |-  ( ph  ->  X  =/=  -u 1
)
2318, 19, 21, 22addneintrd 9776 . . . . . . . 8  |-  ( ph  ->  ( 1  +  X
)  =/=  ( 1  +  -u 1 ) )
24 1pneg1e0 10640 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
2524a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
2623, 25neeqtrd 2749 . . . . . . 7  |-  ( ph  ->  ( 1  +  X
)  =/=  0 )
2720, 18, 20, 26divsubdird 10355 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( 1  / 
( 1  +  X
) ) ) )
2818, 19pncan2d 9924 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  -  1 )  =  X )
2928oveq1d 6285 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  - 
1 )  /  (
1  +  X ) )  =  ( X  /  ( 1  +  X ) ) )
3020, 26dividd 10314 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  X )  /  (
1  +  X ) )  =  1 )
3130oveq1d 6285 . . . . . 6  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  (
1  /  ( 1  +  X ) ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3227, 29, 313eqtr3d 2503 . . . . 5  |-  ( ph  ->  ( X  /  (
1  +  X ) )  =  ( 1  -  ( 1  / 
( 1  +  X
) ) ) )
3332adantr 463 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  =  ( 1  -  (
1  /  ( 1  +  X ) ) ) )
3417, 33breqtrrd 4465 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( X  /  ( 1  +  X ) ) )
35 1m1e0 10600 . . . . . 6  |-  ( 1  -  1 )  =  0
3613simpld 457 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  0  <  ( 1  /  ( 1  +  X ) ) )
3735, 36syl5eqbr 4472 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  1 )  < 
( 1  /  (
1  +  X ) ) )
388, 8, 7, 37ltsub23d 10153 . . . 4  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( 1  -  ( 1  / 
( 1  +  X
) ) )  <  1 )
3933, 38eqbrtrd 4459 . . 3  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  <  1 )
40 0xr 9629 . . . 4  |-  0  e.  RR*
41 1re 9584 . . . . 5  |-  1  e.  RR
4241rexri 9635 . . . 4  |-  1  e.  RR*
43 elioo2 11573 . . . 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 ) ) )
4440, 42, 43mp2an 670 . . 3  |-  ( ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1 )  <->  ( ( X  /  ( 1  +  X ) )  e.  RR  /\  0  < 
( X  /  (
1  +  X ) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
456, 34, 39, 44syl3anbrc 1178 . 2  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
4628adantr 463 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  =  X )
4720adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  CC )
4826adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  =/=  0 )
4947, 48recrecd 10313 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  =  ( 1  +  X ) )
5020, 19, 20, 26divsubdird 10355 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( ( ( 1  +  X
)  /  ( 1  +  X ) )  -  ( X  / 
( 1  +  X
) ) ) )
5118, 19pncand 9923 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  +  X )  -  X
)  =  1 )
5251oveq1d 6285 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  -  X )  /  (
1  +  X ) )  =  ( 1  /  ( 1  +  X ) ) )
5330oveq1d 6285 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1  +  X )  / 
( 1  +  X
) )  -  ( X  /  ( 1  +  X ) ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
5450, 52, 533eqtr3d 2503 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  (
1  +  X ) )  =  ( 1  -  ( X  / 
( 1  +  X
) ) ) )
5554adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  =  ( 1  -  ( X  /  (
1  +  X ) ) ) )
56 1red 9600 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
57 simpr 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )
5857, 44sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( X  /  (
1  +  X ) )  e.  RR  /\  0  <  ( X  / 
( 1  +  X
) )  /\  ( X  /  ( 1  +  X ) )  <  1 ) )
5958simp1d 1006 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  e.  RR )
6056, 59resubcld 9983 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  e.  RR )
6155, 60eqeltrd 2542 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR )
62 0red 9586 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  e.  RR )
6358simp3d 1008 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  <  1 )
6463, 15syl6breqr 4479 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  ( X  /  ( 1  +  X ) )  < 
( 1  -  0 ) )
6559, 56, 62, 64ltsub13d 10154 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  -  ( X  /  ( 1  +  X ) ) ) )
6665, 55breqtrrd 4465 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( 1  /  (
1  +  X ) ) )
6761, 66elrpd 11256 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  e.  RR+ )
6867rprecred 11270 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  /  ( 1  +  X ) ) )  e.  RR )
6949, 68eqeltrrd 2543 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  X )  e.  RR )
7069, 56resubcld 9983 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  +  X
)  -  1 )  e.  RR )
7146, 70eqeltrrd 2543 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR )
72 1p0e1 10644 . . . . 5  |-  ( 1  +  0 )  =  1
7358simp2d 1007 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  ( X  /  (
1  +  X ) ) )
7435, 73syl5eqbr 4472 . . . . . . . . 9  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  1 )  <  ( X  / 
( 1  +  X
) ) )
7556, 56, 59, 74ltsub23d 10153 . . . . . . . 8  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  -  ( X  /  ( 1  +  X ) ) )  <  1 )
7655, 75eqbrtrd 4459 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  /  ( 1  +  X ) )  <  1 )
7767reclt1d 11272 . . . . . . 7  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
( 1  /  (
1  +  X ) )  <  1  <->  1  <  ( 1  / 
( 1  /  (
1  +  X ) ) ) ) )
7876, 77mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  /  (
1  /  ( 1  +  X ) ) ) )
7978, 49breqtrd 4463 . . . . 5  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  1  <  ( 1  +  X
) )
8072, 79syl5eqbr 4472 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
1  +  0 )  <  ( 1  +  X ) )
8162, 71, 56ltadd2d 9727 . . . 4  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  (
0  <  X  <->  ( 1  +  0 )  < 
( 1  +  X
) ) )
8280, 81mpbird 232 . . 3  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  0  <  X )
8371, 82elrpd 11256 . 2  |-  ( (
ph  /\  ( X  /  ( 1  +  X ) )  e.  ( 0 (,) 1
) )  ->  X  e.  RR+ )
8445, 83impbida 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484   RR*cxr 9616    < clt 9617    - cmin 9796   -ucneg 9797    / cdiv 10202   RR+crp 11221   (,)cioo 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-rp 11222  df-ioo 11536
This theorem is referenced by:  angpieqvdlem  23356
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