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Theorem recgt0ii 9662
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 8795 . . . . . 6  |-  1  e.  CC
2 ltplus1.1 . . . . . . 7  |-  A  e.  RR
32recni 8849 . . . . . 6  |-  A  e.  CC
4 ax-1ne0 8806 . . . . . 6  |-  1  =/=  0
5 recgt0i.2 . . . . . . 7  |-  0  <  A
62, 5gt0ne0ii 9309 . . . . . 6  |-  A  =/=  0
71, 3, 4, 6divne0i 9508 . . . . 5  |-  ( 1  /  A )  =/=  0
87necomi 2528 . . . 4  |-  0  =/=  ( 1  /  A
)
9 df-ne 2448 . . . 4  |-  ( 0  =/=  ( 1  /  A )  <->  -.  0  =  ( 1  /  A ) )
108, 9mpbi 199 . . 3  |-  -.  0  =  ( 1  /  A )
11 0lt1 9296 . . . . 5  |-  0  <  1
12 0re 8838 . . . . . 6  |-  0  e.  RR
13 1re 8837 . . . . . 6  |-  1  e.  RR
1412, 13ltnsymi 8937 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
1511, 14ax-mp 8 . . . 4  |-  -.  1  <  0
162, 6rereccli 9525 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1716renegcli 9108 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1817, 2mulgt0i 8951 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
195, 18mpan2 652 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
2016recni 8849 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
2120, 3mulneg1i 9225 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
223, 6recidi 9491 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
233, 20, 22mulcomli 8844 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2423negeqi 9045 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2521, 24eqtri 2303 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2619, 25syl6breq 4062 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
27 lt0neg1 9280 . . . . . 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 9280 . . . . . 6  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
3013, 29ax-mp 8 . . . . 5  |-  ( 1  <  0  <->  0  <  -u 1 )
3126, 28, 303imtr4i 257 . . . 4  |-  ( ( 1  /  A )  <  0  ->  1  <  0 )
3215, 31mto 167 . . 3  |-  -.  (
1  /  A )  <  0
3310, 32pm3.2ni 827 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
34 axlttri 8894 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3512, 16, 34mp2an 653 . 2  |-  ( 0  <  ( 1  /  A )  <->  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 ) )
3633, 35mpbir 200 1  |-  0  <  ( 1  /  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867   -ucneg 9038    / cdiv 9423
This theorem is referenced by:  halfgt0  9932  0.999...  12337  sincos2sgn  12474  rpnnen2lem3  12495  rpnnen2lem4  12496  rpnnen2lem9  12501  pcoass  18522  log2tlbnd  20241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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