MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gt0div Unicode version

Theorem gt0div 9709
Description: Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
Assertion
Ref Expression
gt0div  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B )  ->  (
0  <  A  <->  0  <  ( A  /  B ) ) )

Proof of Theorem gt0div
StepHypRef Expression
1 0re 8925 . . . 4  |-  0  e.  RR
2 ltdiv1 9707 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( 0  <  A  <->  ( 0  /  B )  <  ( A  /  B ) ) )
31, 2mp3an1 1264 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( 0  <  A  <->  ( 0  /  B )  < 
( A  /  B
) ) )
433impb 1147 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B )  ->  (
0  <  A  <->  ( 0  /  B )  < 
( A  /  B
) ) )
5 gt0ne0 9326 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
6 recn 8914 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
7 div0 9539 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 0  /  B
)  =  0 )
86, 7sylan 457 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 0  /  B
)  =  0 )
95, 8syldan 456 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 0  /  B
)  =  0 )
109breq1d 4112 . . 3  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( ( 0  /  B )  <  ( A  /  B )  <->  0  <  ( A  /  B ) ) )
11103adant1 973 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B )  ->  (
( 0  /  B
)  <  ( A  /  B )  <->  0  <  ( A  /  B ) ) )
124, 11bitrd 244 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B )  ->  (
0  <  A  <->  0  <  ( A  /  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4102  (class class class)co 5942   CCcc 8822   RRcr 8823   0cc0 8824    < clt 8954    / cdiv 9510
This theorem is referenced by:  divgt0  9711  halfpos2  10030  gt0divd  10512  dvferm1lem  19429  dvferm2lem  19431  dvgt0  19449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511
  Copyright terms: Public domain W3C validator