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Theorem ledivp1 9674
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 732 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  e.  RR )
2 peano2re 9001 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
32ad2antrl 708 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  RR )
4 simpll 730 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  RR )
5 ltp1 9610 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  <  ( B  +  1 ) )
6 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
7 lelttr 8928 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  ( B  +  1 )  e.  RR )  -> 
( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
86, 7mp3an1 1264 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( B  +  1
)  e.  RR )  ->  ( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
92, 8mpdan 649 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
105, 9mpan2d 655 . . . . . . . 8  |-  ( B  e.  RR  ->  (
0  <_  B  ->  0  <  ( B  + 
1 ) ) )
1110imp 418 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
0  <  ( B  +  1 ) )
1211gt0ne0d 9353 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  =/=  0 )
1312adantl 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  =/=  0 )
144, 3, 13redivcld 9604 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  /  ( B  + 
1 ) )  e.  RR )
152adantr 451 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  e.  RR )
1615, 11jca 518 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )
17 divge0 9641 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1816, 17sylan2 460 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1914, 18jca 518 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  e.  RR  /\  0  <_ 
( A  /  ( B  +  1 ) ) ) )
20 lep1 9611 . . . 4  |-  ( B  e.  RR  ->  B  <_  ( B  +  1 ) )
2120ad2antrl 708 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  <_  ( B  +  1 ) )
22 lemul2a 9627 . . 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 ) ) )
231, 3, 19, 21, 22syl31anc 1185 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  (
( A  /  ( B  +  1 ) )  x.  ( B  +  1 ) ) )
24 recn 8843 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2524ad2antrr 706 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  CC )
262recnd 8877 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  CC )
2726ad2antrl 708 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  CC )
2825, 27, 13divcan1d 9553 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  ( B  +  1 ) )  =  A )
2923, 28breqtrd 4063 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 358    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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