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Theorem recgt0ii 9678
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 8811 . . . . . 6  |-  1  e.  CC
2 ltplus1.1 . . . . . . 7  |-  A  e.  RR
32recni 8865 . . . . . 6  |-  A  e.  CC
4 ax-1ne0 8822 . . . . . 6  |-  1  =/=  0
5 recgt0i.2 . . . . . . 7  |-  0  <  A
62, 5gt0ne0ii 9325 . . . . . 6  |-  A  =/=  0
71, 3, 4, 6divne0i 9524 . . . . 5  |-  ( 1  /  A )  =/=  0
87necomi 2541 . . . 4  |-  0  =/=  ( 1  /  A
)
9 df-ne 2461 . . . 4  |-  ( 0  =/=  ( 1  /  A )  <->  -.  0  =  ( 1  /  A ) )
108, 9mpbi 199 . . 3  |-  -.  0  =  ( 1  /  A )
11 0lt1 9312 . . . . 5  |-  0  <  1
12 0re 8854 . . . . . 6  |-  0  e.  RR
13 1re 8853 . . . . . 6  |-  1  e.  RR
1412, 13ltnsymi 8953 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
1511, 14ax-mp 8 . . . 4  |-  -.  1  <  0
162, 6rereccli 9541 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1716renegcli 9124 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1817, 2mulgt0i 8967 . . . . . . 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 8865 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
2120, 3mulneg1i 9241 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
223, 6recidi 9507 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
233, 20, 22mulcomli 8860 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2423negeqi 9061 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2521, 24eqtri 2316 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2619, 25syl6breq 4078 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
27 lt0neg1 9296 . . . . . 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 9296 . . . . . 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 8910 . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883   -ucneg 9054    / cdiv 9439
This theorem is referenced by:  halfgt0  9948  0.999...  12353  sincos2sgn  12490  rpnnen2lem3  12511  rpnnen2lem4  12512  rpnnen2lem9  12517  pcoass  18538  log2tlbnd  20257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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