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

Theorem recgt0 10165
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 457 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
21recnd 9404 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
3 gt0ne0 9796 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
42, 3recne0d 10093 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  =/=  0 )
54necomd 2690 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  =/=  ( 1  /  A ) )
65neneqd 2619 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  0  =  (
1  /  A ) )
7 0lt1 9854 . . . . 5  |-  0  <  1
8 0re 9378 . . . . . 6  |-  0  e.  RR
9 1re 9377 . . . . . 6  |-  1  e.  RR
108, 9ltnsymi 9485 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
117, 10ax-mp 5 . . . 4  |-  -.  1  <  0
12 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  RR )
133adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  =/=  0 )
1412, 13rereccld 10150 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1514renegcld 9767 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( 1  /  A )  e.  RR )
16 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  <  0 )
171, 3rereccld 10150 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
1817adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1918lt0neg1d 9901 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) ) )
2016, 19mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u ( 1  /  A
) )
21 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  A )
2215, 12, 20, 21mulgt0d 9518 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
232adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  CC )
2423, 13reccld 10092 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  CC )
2524, 23mulneg1d 9789 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u ( ( 1  /  A )  x.  A ) )
2623, 13recid2d 10095 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  x.  A )  =  1 )
2726negeqd 9596 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( ( 1  /  A )  x.  A )  = 
-u 1 )
2825, 27eqtrd 2470 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u 1 )
2922, 28breqtrd 4311 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u 1 )
30 1red 9393 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  e.  RR )
3130lt0neg1d 9901 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  <  0  <->  0  <  -u 1 ) )
3229, 31mpbird 232 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  <  0 )
3332ex 434 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  0  ->  1  <  0 ) )
3411, 33mtoi 178 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 1  /  A
)  <  0 )
35 ioran 490 . . 3  |-  ( -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
)  <->  ( -.  0  =  ( 1  /  A )  /\  -.  ( 1  /  A
)  <  0 ) )
366, 34, 35sylanbrc 664 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) )
37 axlttri 9438 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
388, 17, 37sylancr 663 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) ) )
3936, 38mpbird 232 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    < clt 9410   -ucneg 9588    / cdiv 9985
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986
This theorem is referenced by:  prodgt0  10166  ltdiv1  10185  ltrec1  10211  lerec2  10212  lediv12a  10217  recgt1i  10221  recreclt  10223  recgt0i  10229  recgt0d  10259  nnrecgt0  10351  nnrecl  10569  resqrex  12732  leopmul  25506  cdj1i  25805
  Copyright terms: Public domain W3C validator