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Theorem recgt0 9568
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 445 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
21recnd 8829 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
3 gt0ne0 9207 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
42, 3recne0d 9498 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  =/=  0 )
54necomd 2504 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  =/=  ( 1  /  A ) )
65neneqd 2437 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  0  =  (
1  /  A ) )
7 0lt1 9264 . . . . 5  |-  0  <  1
8 0re 8806 . . . . . 6  |-  0  e.  RR
9 1re 8805 . . . . . 6  |-  1  e.  RR
108, 9ltnsymi 8905 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
117, 10ax-mp 10 . . . 4  |-  -.  1  <  0
12 simpll 733 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  RR )
133adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  =/=  0 )
1412, 13rereccld 9555 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1514renegcld 9178 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( 1  /  A )  e.  RR )
16 simpr 449 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  <  0 )
171, 3rereccld 9555 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
1817adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1918lt0neg1d 9310 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) ) )
2016, 19mpbid 203 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u ( 1  /  A
) )
21 simplr 734 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  A )
2215, 12, 20, 21mulgt0d 8939 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
232adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  CC )
2423, 13reccld 9497 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  CC )
2524, 23mulneg1d 9200 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u ( ( 1  /  A )  x.  A ) )
2623, 13recid2d 9500 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  x.  A )  =  1 )
2726negeqd 9014 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( ( 1  /  A )  x.  A )  = 
-u 1 )
2825, 27eqtrd 2290 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u 1 )
2922, 28breqtrd 4021 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u 1 )
309a1i 12 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  e.  RR )
3130lt0neg1d 9310 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  <  0  <->  0  <  -u 1 ) )
3229, 31mpbird 225 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  <  0 )
3332ex 425 . . . 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 478 . . 3  |-  ( -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
)  <->  ( -.  0  =  ( 1  /  A )  /\  -.  ( 1  /  A
)  <  0 ) )
366, 34, 35sylanbrc 648 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) )
37 axlttri 8862 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
388, 17, 37sylancr 647 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  -.  ( 0  =  ( 1  /  A )  \/  ( 1  /  A )  <  0
) ) )
3936, 38mpbird 225 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    x. cmul 8710    < clt 8835   -ucneg 9006    / cdiv 9391
This theorem is referenced by:  prodgt0  9569  ltdiv1  9588  ltrec1  9611  lerec2  9612  lediv12a  9617  recgt1i  9621  recreclt  9623  recgt0i  9629  recgt0d  9659  nnrecgt0  9751  nnrecl  9930  resqrex  11701  leopmul  22674  cdj1i  22973
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392
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