MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recgt0 Unicode version

Theorem recgt0 9843
Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
recgt0  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )

Proof of Theorem recgt0
StepHypRef Expression
1 simpl 444 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
21recnd 9103 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
3 gt0ne0 9482 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
42, 3recne0d 9773 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  =/=  0 )
54necomd 2681 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  =/=  ( 1  /  A ) )
65neneqd 2614 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  0  =  (
1  /  A ) )
7 0lt1 9539 . . . . 5  |-  0  <  1
8 0re 9080 . . . . . 6  |-  0  e.  RR
9 1re 9079 . . . . . 6  |-  1  e.  RR
108, 9ltnsymi 9181 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
117, 10ax-mp 8 . . . 4  |-  -.  1  <  0
12 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  RR )
133adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  =/=  0 )
1412, 13rereccld 9830 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1514renegcld 9453 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( 1  /  A )  e.  RR )
16 simpr 448 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  <  0 )
171, 3rereccld 9830 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
1817adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1918lt0neg1d 9585 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) ) )
2016, 19mpbid 202 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u ( 1  /  A
) )
21 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  A )
2215, 12, 20, 21mulgt0d 9214 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
232adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  CC )
2423, 13reccld 9772 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  CC )
2524, 23mulneg1d 9475 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u ( ( 1  /  A )  x.  A ) )
2623, 13recid2d 9775 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  x.  A )  =  1 )
2726negeqd 9289 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( ( 1  /  A )  x.  A )  = 
-u 1 )
2825, 27eqtrd 2467 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u 1 )
2922, 28breqtrd 4228 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u 1 )
309a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  e.  RR )
3130lt0neg1d 9585 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  <  0  <->  0  <  -u 1 ) )
3229, 31mpbird 224 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  <  0 )
3332ex 424 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  0  ->  1  <  0 ) )
3411, 33mtoi 171 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 1  /  A
)  <  0 )
35 ioran 477 . . 3  |-  ( -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
)  <->  ( -.  0  =  ( 1  /  A )  /\  -.  ( 1  /  A
)  <  0 ) )
366, 34, 35sylanbrc 646 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) )
37 axlttri 9136 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
388, 17, 37sylancr 645 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) ) )
3936, 38mpbird 224 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984    < clt 9109   -ucneg 9281    / cdiv 9666
This theorem is referenced by:  prodgt0  9844  ltdiv1  9863  ltrec1  9886  lerec2  9887  lediv12a  9892  recgt1i  9896  recreclt  9898  recgt0i  9904  recgt0d  9934  nnrecgt0  10026  nnrecl  10208  resqrex  12044  leopmul  23625  cdj1i  23924
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 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
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-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667
  Copyright terms: Public domain W3C validator