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Theorem recgt0ii 10353
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 9455 . . . . 5  |-  1  e.  CC
2 ltplus1.1 . . . . . 6  |-  A  e.  RR
32recni 9513 . . . . 5  |-  A  e.  CC
4 ax-1ne0 9466 . . . . 5  |-  1  =/=  0
5 recgt0i.2 . . . . . 6  |-  0  <  A
62, 5gt0ne0ii 9991 . . . . 5  |-  A  =/=  0
71, 3, 4, 6divne0i 10194 . . . 4  |-  ( 1  /  A )  =/=  0
87nesymi 2725 . . 3  |-  -.  0  =  ( 1  /  A )
9 0lt1 9977 . . . . 5  |-  0  <  1
10 0re 9501 . . . . . 6  |-  0  e.  RR
11 1re 9500 . . . . . 6  |-  1  e.  RR
1210, 11ltnsymi 9608 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
139, 12ax-mp 5 . . . 4  |-  -.  1  <  0
142, 6rereccli 10211 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1514renegcli 9785 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1615, 2mulgt0i 9621 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
175, 16mpan2 671 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
1814recni 9513 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
1918, 3mulneg1i 9905 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
203, 6recidi 10177 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
213, 18, 20mulcomli 9508 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2221negeqi 9718 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2319, 22eqtri 2483 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2417, 23syl6breq 4442 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
25 lt0neg1 9960 . . . . . 6  |-  ( ( 1  /  A )  e.  RR  ->  (
( 1  /  A
)  <  0  <->  0  <  -u ( 1  /  A
) ) )
2614, 25ax-mp 5 . . . . 5  |-  ( ( 1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) )
27 lt0neg1 9960 . . . . . 6  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
2811, 27ax-mp 5 . . . . 5  |-  ( 1  <  0  <->  0  <  -u 1 )
2924, 26, 283imtr4i 266 . . . 4  |-  ( ( 1  /  A )  <  0  ->  1  <  0 )
3013, 29mto 176 . . 3  |-  -.  (
1  /  A )  <  0
318, 30pm3.2ni 850 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
32 axlttri 9561 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3310, 14, 32mp2an 672 . 2  |-  ( 0  <  ( 1  /  A )  <->  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 ) )
3431, 33mpbir 209 1  |-  0  <  ( 1  /  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758   class class class wbr 4403  (class class class)co 6203   RRcr 9396   0cc0 9397   1c1 9398    x. cmul 9402    < clt 9533   -ucneg 9711    / cdiv 10108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109
This theorem is referenced by:  halfgt0  10657  0.999...  13463  sincos2sgn  13600  rpnnen2lem3  13621  rpnnen2lem4  13622  rpnnen2lem9  13627  pcoass  20738  log2tlbnd  22483  stoweidlem34  30000  stoweidlem59  30025
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