ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ledivp1 Unicode version

Theorem ledivp1 8877
Description: Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
Assertion
Ref Expression
ledivp1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  A
)

Proof of Theorem ledivp1
StepHypRef Expression
1 simprl 529 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  e.  RR )
2 peano2re 8110 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
31, 2syl 14 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  RR )
4 simpll 527 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  RR )
5 0red 7975 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  e.  RR )
6 simprr 531 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  B )
7 ltp1 8818 . . . . . . . 8  |-  ( B  e.  RR  ->  B  <  ( B  +  1 ) )
81, 7syl 14 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  <  ( B  +  1 ) )
95, 1, 3, 6, 8lelttrd 8099 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <  ( B  +  1 ) )
103, 9gt0ap0d 8603 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 ) #  0 )
114, 3, 10redivclapd 8809 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  /  ( B  + 
1 ) )  e.  RR )
12 simpl 109 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  e.  RR  /\  0  <_  A ) )
13 divge0 8847 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1412, 3, 9, 13syl12anc 1246 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1511, 14jca 306 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  e.  RR  /\  0  <_ 
( A  /  ( B  +  1 ) ) ) )
161, 3, 8ltled 8093 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  <_  ( B  +  1 ) )
17 lemul2a 8833 . . 3  |-  ( ( ( B  e.  RR  /\  ( B  +  1 )  e.  RR  /\  ( ( A  / 
( B  +  1 ) )  e.  RR  /\  0  <_  ( A  /  ( B  + 
1 ) ) ) )  /\  B  <_ 
( B  +  1 ) )  ->  (
( A  /  ( B  +  1 ) )  x.  B )  <_  ( ( A  /  ( B  + 
1 ) )  x.  ( B  +  1 ) ) )
181, 3, 15, 16, 17syl31anc 1251 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  (
( A  /  ( B  +  1 ) )  x.  ( B  +  1 ) ) )
194recnd 8003 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  CC )
203recnd 8003 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  CC )
2119, 20, 10divcanap1d 8765 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  ( B  +  1 ) )  =  A )
2218, 21breqtrd 4043 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2159   class class class wbr 4017  (class class class)co 5890   RRcr 7827   0cc0 7828   1c1 7829    + caddc 7831    x. cmul 7833    < clt 8009    <_ cle 8010    / cdiv 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator