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Theorem recgt0ii 9848
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 8981 . . . . . 6  |-  1  e.  CC
2 ltplus1.1 . . . . . . 7  |-  A  e.  RR
32recni 9035 . . . . . 6  |-  A  e.  CC
4 ax-1ne0 8992 . . . . . 6  |-  1  =/=  0
5 recgt0i.2 . . . . . . 7  |-  0  <  A
62, 5gt0ne0ii 9495 . . . . . 6  |-  A  =/=  0
71, 3, 4, 6divne0i 9694 . . . . 5  |-  ( 1  /  A )  =/=  0
87necomi 2632 . . . 4  |-  0  =/=  ( 1  /  A
)
9 df-ne 2552 . . . 4  |-  ( 0  =/=  ( 1  /  A )  <->  -.  0  =  ( 1  /  A ) )
108, 9mpbi 200 . . 3  |-  -.  0  =  ( 1  /  A )
11 0lt1 9482 . . . . 5  |-  0  <  1
12 0re 9024 . . . . . 6  |-  0  e.  RR
13 1re 9023 . . . . . 6  |-  1  e.  RR
1412, 13ltnsymi 9123 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
1511, 14ax-mp 8 . . . 4  |-  -.  1  <  0
162, 6rereccli 9711 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1716renegcli 9294 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1817, 2mulgt0i 9137 . . . . . . 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 9035 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
2120, 3mulneg1i 9411 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
223, 6recidi 9677 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
233, 20, 22mulcomli 9030 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2423negeqi 9231 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2521, 24eqtri 2407 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2619, 25syl6breq 4192 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
27 lt0neg1 9466 . . . . . 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 9466 . . . . . 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 9080 . . 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 1717    =/= wne 2550   class class class wbr 4153  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    < clt 9053   -ucneg 9224    / cdiv 9609
This theorem is referenced by:  halfgt0  10120  0.999...  12585  sincos2sgn  12722  rpnnen2lem3  12743  rpnnen2lem4  12744  rpnnen2lem9  12749  pcoass  18920  log2tlbnd  20652  stoweidlem34  27451  stoweidlem59  27476
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610
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