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Theorem ledivp1 10405
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 755 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  e.  RR )
2 peano2re 9705 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
32ad2antrl 726 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  RR )
4 simpll 752 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  RR )
5 ltp1 10339 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  <  ( B  +  1 ) )
6 0re 9544 . . . . . . . . . . 11  |-  0  e.  RR
7 lelttr 9624 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  ( B  +  1 )  e.  RR )  -> 
( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
86, 7mp3an1 1311 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( B  +  1
)  e.  RR )  ->  ( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
92, 8mpdan 666 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
105, 9mpan2d 672 . . . . . . . 8  |-  ( B  e.  RR  ->  (
0  <_  B  ->  0  <  ( B  + 
1 ) ) )
1110imp 427 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
0  <  ( B  +  1 ) )
1211gt0ne0d 10075 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  =/=  0 )
1312adantl 464 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  =/=  0 )
144, 3, 13redivcld 10331 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  /  ( B  + 
1 ) )  e.  RR )
152adantr 463 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  e.  RR )
1615, 11jca 530 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )
17 divge0 10370 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1816, 17sylan2 472 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1914, 18jca 530 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  e.  RR  /\  0  <_ 
( A  /  ( B  +  1 ) ) ) )
20 lep1 10340 . . . 4  |-  ( B  e.  RR  ->  B  <_  ( B  +  1 ) )
2120ad2antrl 726 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  <_  ( B  +  1 ) )
22 lemul2a 10356 . . 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 1231 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  (
( A  /  ( B  +  1 ) )  x.  ( B  +  1 ) ) )
24 recn 9530 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2524ad2antrr 724 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  CC )
262recnd 9570 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  CC )
2726ad2antrl 726 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  CC )
2825, 27, 13divcan1d 10280 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  ( B  +  1 ) )  =  A )
2923, 28breqtrd 4416 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840    =/= wne 2596   class class class wbr 4392  (class class class)co 6232   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445    < clt 9576    <_ cle 9577    / cdiv 10165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166
This theorem is referenced by: (None)
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