MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recgt0ii Structured version   Unicode version

Theorem recgt0ii 10514
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 9599 . . . . 5  |-  1  e.  CC
2 ltplus1.1 . . . . . 6  |-  A  e.  RR
32recni 9657 . . . . 5  |-  A  e.  CC
4 ax-1ne0 9610 . . . . 5  |-  1  =/=  0
5 recgt0i.2 . . . . . 6  |-  0  <  A
62, 5gt0ne0ii 10152 . . . . 5  |-  A  =/=  0
71, 3, 4, 6divne0i 10357 . . . 4  |-  ( 1  /  A )  =/=  0
87nesymi 2698 . . 3  |-  -.  0  =  ( 1  /  A )
9 0lt1 10138 . . . . 5  |-  0  <  1
10 0re 9645 . . . . . 6  |-  0  e.  RR
11 1re 9644 . . . . . 6  |-  1  e.  RR
1210, 11ltnsymi 9755 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
139, 12ax-mp 5 . . . 4  |-  -.  1  <  0
142, 6rereccli 10374 . . . . . . . . 9  |-  ( 1  /  A )  e.  RR
1514renegcli 9937 . . . . . . . 8  |-  -u (
1  /  A )  e.  RR
1615, 2mulgt0i 9769 . . . . . . 7  |-  ( ( 0  <  -u (
1  /  A )  /\  0  <  A
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
175, 16mpan2 676 . . . . . 6  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  ( -u ( 1  /  A )  x.  A ) )
1814recni 9657 . . . . . . . 8  |-  ( 1  /  A )  e.  CC
1918, 3mulneg1i 10066 . . . . . . 7  |-  ( -u ( 1  /  A
)  x.  A )  =  -u ( ( 1  /  A )  x.  A )
203, 6recidi 10340 . . . . . . . . 9  |-  ( A  x.  ( 1  /  A ) )  =  1
213, 18, 20mulcomli 9652 . . . . . . . 8  |-  ( ( 1  /  A )  x.  A )  =  1
2221negeqi 9870 . . . . . . 7  |-  -u (
( 1  /  A
)  x.  A )  =  -u 1
2319, 22eqtri 2452 . . . . . 6  |-  ( -u ( 1  /  A
)  x.  A )  =  -u 1
2417, 23syl6breq 4461 . . . . 5  |-  ( 0  <  -u ( 1  /  A )  ->  0  <  -u 1 )
25 lt0neg1 10122 . . . . . 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 10122 . . . . . 6  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
2811, 27ax-mp 5 . . . . 5  |-  ( 1  <  0  <->  0  <  -u 1 )
2924, 26, 283imtr4i 270 . . . 4  |-  ( ( 1  /  A )  <  0  ->  1  <  0 )
3013, 29mto 180 . . 3  |-  -.  (
1  /  A )  <  0
318, 30pm3.2ni 863 . 2  |-  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 )
32 axlttri 9707 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  -.  ( 0  =  ( 1  /  A )  \/  (
1  /  A )  <  0 ) ) )
3310, 14, 32mp2an 677 . 2  |-  ( 0  <  ( 1  /  A )  <->  -.  (
0  =  ( 1  /  A )  \/  ( 1  /  A
)  <  0 ) )
3431, 33mpbir 213 1  |-  0  <  ( 1  /  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    = wceq 1438    e. wcel 1869   class class class wbr 4421  (class class class)co 6303   RRcr 9540   0cc0 9541   1c1 9542    x. cmul 9546    < clt 9677   -ucneg 9863    / cdiv 10271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272
This theorem is referenced by:  halfgt0  10832  0.999...  13930  sincos2sgn  14241  rpnnen2lem3  14262  rpnnen2lem4  14263  rpnnen2lem9  14268  pcoass  22047  log2tlbnd  23863  stoweidlem34  37759  stoweidlem59  37784
  Copyright terms: Public domain W3C validator