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Theorem recgt0ii 10226
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 9328 . . . . 5  |-  1  e.  CC
2 ltplus1.1 . . . . . 6  |-  A  e.  RR
32recni 9386 . . . . 5  |-  A  e.  CC
4 ax-1ne0 9339 . . . . 5  |-  1  =/=  0
5 recgt0i.2 . . . . . 6  |-  0  <  A
62, 5gt0ne0ii 9864 . . . . 5  |-  A  =/=  0
71, 3, 4, 6divne0i 10067 . . . 4  |-  ( 1  /  A )  =/=  0
87nesymi 2638 . . 3  |-  -.  0  =  ( 1  /  A )
9 0lt1 9850 . . . . 5  |-  0  <  1
10 0re 9374 . . . . . 6  |-  0  e.  RR
11 1re 9373 . . . . . 6  |-  1  e.  RR
1210, 11ltnsymi 9481 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
139, 12ax-mp 5 . . . 4  |-  -.  1  <  0
142, 6rereccli 10084 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1514renegcli 9658 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1615, 2mulgt0i 9494 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
175, 16mpan2 664 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
1814recni 9386 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
1918, 3mulneg1i 9778 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
203, 6recidi 10050 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
213, 18, 20mulcomli 9381 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2221negeqi 9591 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2319, 22eqtri 2453 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2417, 23syl6breq 4319 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
25 lt0neg1 9833 . . . . . 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 9833 . . . . . 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 843 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
32 axlttri 9434 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3310, 14, 32mp2an 665 . 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 1362    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9269   0cc0 9270   1c1 9271    x. cmul 9275    < clt 9406   -ucneg 9584    / cdiv 9981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982
This theorem is referenced by:  halfgt0  10530  0.999...  13324  sincos2sgn  13461  rpnnen2lem3  13482  rpnnen2lem4  13483  rpnnen2lem9  13488  pcoass  20438  log2tlbnd  22225  stoweidlem34  29675  stoweidlem59  29700
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