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Theorem recgt0ii 9630
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 8763 . . . . . 6  |-  1  e.  CC
2 ltplus1.1 . . . . . . 7  |-  A  e.  RR
32recni 8817 . . . . . 6  |-  A  e.  CC
4 ax-1ne0 8774 . . . . . 6  |-  1  =/=  0
5 recgt0i.2 . . . . . . 7  |-  0  <  A
62, 5gt0ne0ii 9277 . . . . . 6  |-  A  =/=  0
71, 3, 4, 6divne0i 9476 . . . . 5  |-  ( 1  /  A )  =/=  0
87necomi 2503 . . . 4  |-  0  =/=  ( 1  /  A
)
9 df-ne 2423 . . . 4  |-  ( 0  =/=  ( 1  /  A )  <->  -.  0  =  ( 1  /  A ) )
108, 9mpbi 201 . . 3  |-  -.  0  =  ( 1  /  A )
11 0lt1 9264 . . . . 5  |-  0  <  1
12 0re 8806 . . . . . 6  |-  0  e.  RR
13 1re 8805 . . . . . 6  |-  1  e.  RR
1412, 13ltnsymi 8905 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
1511, 14ax-mp 10 . . . 4  |-  -.  1  <  0
162, 6rereccli 9493 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1716renegcli 9076 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1817, 2mulgt0i 8919 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
195, 18mpan2 655 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
2016recni 8817 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
2120, 3mulneg1i 9193 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
223, 6recidi 9459 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
233, 20, 22mulcomli 8812 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2423negeqi 9013 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2521, 24eqtri 2278 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2619, 25syl6breq 4036 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
27 lt0neg1 9248 . . . . . 6  |-  ( ( 1  /  A )  e.  RR  ->  (
( 1  /  A
)  <  0  <->  0  <  -u ( 1  /  A
) ) )
2816, 27ax-mp 10 . . . . 5  |-  ( ( 1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) )
29 lt0neg1 9248 . . . . . 6  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
3013, 29ax-mp 10 . . . . 5  |-  ( 1  <  0  <->  0  <  -u 1 )
3126, 28, 303imtr4i 259 . . . 4  |-  ( ( 1  /  A )  <  0  ->  1  <  0 )
3215, 31mto 169 . . 3  |-  -.  (
1  /  A )  <  0
3310, 32pm3.2ni 830 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
34 axlttri 8862 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3512, 16, 34mp2an 656 . 2  |-  ( 0  <  ( 1  /  A )  <->  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 ) )
3633, 35mpbir 202 1  |-  0  <  ( 1  /  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997  (class class class)co 5792   RRcr 8704   0cc0 8705   1c1 8706    x. cmul 8710    < clt 8835   -ucneg 9006    / cdiv 9391
This theorem is referenced by:  halfgt0  9899  0.999...  12299  sincos2sgn  12436  rpnnen2lem3  12457  rpnnen2lem4  12458  rpnnen2lem9  12463  pcoass  18484  log2tlbnd  20203
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392
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