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Theorem recgt0ii 9872
Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
ltplus1.1  |-  A  e.  RR
recgt0i.2  |-  0  <  A
Assertion
Ref Expression
recgt0ii  |-  0  <  ( 1  /  A
)

Proof of Theorem recgt0ii
StepHypRef Expression
1 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
2 ltplus1.1 . . . . . . 7  |-  A  e.  RR
32recni 9058 . . . . . 6  |-  A  e.  CC
4 ax-1ne0 9015 . . . . . 6  |-  1  =/=  0
5 recgt0i.2 . . . . . . 7  |-  0  <  A
62, 5gt0ne0ii 9519 . . . . . 6  |-  A  =/=  0
71, 3, 4, 6divne0i 9718 . . . . 5  |-  ( 1  /  A )  =/=  0
87necomi 2649 . . . 4  |-  0  =/=  ( 1  /  A
)
9 df-ne 2569 . . . 4  |-  ( 0  =/=  ( 1  /  A )  <->  -.  0  =  ( 1  /  A ) )
108, 9mpbi 200 . . 3  |-  -.  0  =  ( 1  /  A )
11 0lt1 9506 . . . . 5  |-  0  <  1
12 0re 9047 . . . . . 6  |-  0  e.  RR
13 1re 9046 . . . . . 6  |-  1  e.  RR
1412, 13ltnsymi 9148 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
1511, 14ax-mp 8 . . . 4  |-  -.  1  <  0
162, 6rereccli 9735 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1716renegcli 9318 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1817, 2mulgt0i 9161 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
195, 18mpan2 653 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
2016recni 9058 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
2120, 3mulneg1i 9435 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
223, 6recidi 9701 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
233, 20, 22mulcomli 9053 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2423negeqi 9255 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2521, 24eqtri 2424 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2619, 25syl6breq 4211 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
27 lt0neg1 9490 . . . . . 6  |-  ( ( 1  /  A )  e.  RR  ->  (
( 1  /  A
)  <  0  <->  0  <  -u ( 1  /  A
) ) )
2816, 27ax-mp 8 . . . . 5  |-  ( ( 1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) )
29 lt0neg1 9490 . . . . . 6  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
3013, 29ax-mp 8 . . . . 5  |-  ( 1  <  0  <->  0  <  -u 1 )
3126, 28, 303imtr4i 258 . . . 4  |-  ( ( 1  /  A )  <  0  ->  1  <  0 )
3215, 31mto 169 . . 3  |-  -.  (
1  /  A )  <  0
3310, 32pm3.2ni 828 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
34 axlttri 9103 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3512, 16, 34mp2an 654 . 2  |-  ( 0  <  ( 1  /  A )  <->  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 ) )
3633, 35mpbir 201 1  |-  0  <  ( 1  /  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076   -ucneg 9248    / cdiv 9633
This theorem is referenced by:  halfgt0  10144  0.999...  12613  sincos2sgn  12750  rpnnen2lem3  12771  rpnnen2lem4  12772  rpnnen2lem9  12777  pcoass  19002  log2tlbnd  20738  stoweidlem34  27650  stoweidlem59  27675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634
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