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Theorem ledivp1 10348
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 9656 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
32ad2antrl 727 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  RR )
4 simpll 753 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  RR )
5 ltp1 10281 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  <  ( B  +  1 ) )
6 0re 9500 . . . . . . . . . . 11  |-  0  e.  RR
7 lelttr 9579 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  ( B  +  1 )  e.  RR )  -> 
( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
86, 7mp3an1 1302 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( B  +  1
)  e.  RR )  ->  ( ( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
92, 8mpdan 668 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
( 0  <_  B  /\  B  <  ( B  +  1 ) )  ->  0  <  ( B  +  1 ) ) )
105, 9mpan2d 674 . . . . . . . 8  |-  ( B  e.  RR  ->  (
0  <_  B  ->  0  <  ( B  + 
1 ) ) )
1110imp 429 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
0  <  ( B  +  1 ) )
1211gt0ne0d 10018 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  =/=  0 )
1312adantl 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  =/=  0 )
144, 3, 13redivcld 10273 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  /  ( B  + 
1 ) )  e.  RR )
152adantr 465 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( B  +  1 )  e.  RR )
1615, 11jca 532 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B )  -> 
( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )
17 divge0 10312 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( ( B  + 
1 )  e.  RR  /\  0  <  ( B  +  1 ) ) )  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1816, 17sylan2 474 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  /  ( B  +  1 ) ) )
1914, 18jca 532 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  e.  RR  /\  0  <_ 
( A  /  ( B  +  1 ) ) ) )
20 lep1 10282 . . . 4  |-  ( B  e.  RR  ->  B  <_  ( B  +  1 ) )
2120ad2antrl 727 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  <_  ( B  +  1 ) )
22 lemul2a 10298 . . 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 1222 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  B )  <_  (
( A  /  ( B  +  1 ) )  x.  ( B  +  1 ) ) )
24 recn 9486 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2524ad2antrr 725 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  CC )
262recnd 9526 . . . 4  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  CC )
2726ad2antrl 727 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  +  1 )  e.  CC )
2825, 27, 13divcan1d 10222 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  /  ( B  + 
1 ) )  x.  ( B  +  1 ) )  =  A )
2923, 28breqtrd 4427 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 369    e. wcel 1758    =/= wne 2648   class class class wbr 4403  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    < clt 9532    <_ cle 9533    / cdiv 10107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108
This theorem is referenced by: (None)
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