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Theorem ltdifltdiv 11765
Description: If the dividend of a division is less than the difference between a real number and the divisor, the floor function of the division plus 1 is less than the division of the real number by the divisor. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
Assertion
Ref Expression
ltdifltdiv  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  <  ( C  -  B )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  < 
( C  /  B
) ) )

Proof of Theorem ltdifltdiv
StepHypRef Expression
1 refldivcl 11756 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
2 peano2re 9629 . . . . . 6  |-  ( ( |_ `  ( A  /  B ) )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
31, 2syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( |_ `  ( A  /  B
) )  +  1 )  e.  RR )
433adant3 1008 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
54adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
6 rerpdivcl 11105 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
7 peano2re 9629 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  +  1 )  e.  RR )
86, 7syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  +  1 )  e.  RR )
983adant3 1008 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  +  1 )  e.  RR )
109adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( A  /  B
)  +  1 )  e.  RR )
11 rerpdivcl 11105 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR+ )  -> 
( C  /  B
)  e.  RR )
1211ancoms 453 . . . . 5  |-  ( ( B  e.  RR+  /\  C  e.  RR )  ->  ( C  /  B )  e.  RR )
13123adant1 1006 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( C  /  B )  e.  RR )
1413adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( C  /  B )  e.  RR )
1513adant3 1008 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
1615adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
1763adant3 1008 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
1817adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  /  B )  e.  RR )
19 1red 9488 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  1  e.  RR )
20 3simpa 985 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
2120adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
22 fldivle 11762 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
2321, 22syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
2416, 18, 19, 23leadd1dd 10040 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  <_ 
( ( A  /  B )  +  1 ) )
25 rpre 11084 . . . . . . 7  |-  ( B  e.  RR+  ->  B  e.  RR )
26 ltaddsub 9900 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
2725, 26syl3an2 1253 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
2827biimpar 485 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  +  B )  <  C )
29 recn 9459 . . . . . . . . . . 11  |-  ( ( A  /  B )  e.  RR  ->  ( A  /  B )  e.  CC )
306, 29syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
31303adant3 1008 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  /  B )  e.  CC )
32 ax-1cn 9427 . . . . . . . . . 10  |-  1  e.  CC
3332a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  1  e.  CC )
34 rpcn 11086 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  B  e.  CC )
35343ad2ant2 1010 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  CC )
3631, 33, 35adddird 9498 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( ( ( A  /  B )  x.  B )  +  ( 1  x.  B
) ) )
37 recn 9459 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
38373ad2ant1 1009 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  A  e.  CC )
39 rpne0 11093 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  B  =/=  0 )
40393ad2ant2 1010 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  =/=  0 )
4138, 35, 40divcan1d 10195 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  x.  B )  =  A )
4234mulid2d 9491 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  ( 1  x.  B )  =  B )
43423ad2ant2 1010 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
1  x.  B )  =  B )
4441, 43oveq12d 6194 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  x.  B
)  +  ( 1  x.  B ) )  =  ( A  +  B ) )
4536, 44eqtrd 2490 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( A  +  B ) )
46 recn 9459 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
47463ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  C  e.  CC )
4847, 35, 40divcan1d 10195 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( C  /  B
)  x.  B )  =  C )
4945, 48breq12d 4389 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( ( A  /  B )  +  1 )  x.  B
)  <  ( ( C  /  B )  x.  B )  <->  ( A  +  B )  <  C
) )
5049adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( ( ( A  /  B )  +  1 )  x.  B
)  <  ( ( C  /  B )  x.  B )  <->  ( A  +  B )  <  C
) )
5128, 50mpbird 232 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( ( A  /  B )  +  1 )  x.  B )  <  ( ( C  /  B )  x.  B ) )
5217, 7syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  +  1 )  e.  RR )
53 simp2 989 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  RR+ )
5452, 13, 53ltmul1d 11151 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  <  ( C  /  B )  <->  ( (
( A  /  B
)  +  1 )  x.  B )  < 
( ( C  /  B )  x.  B
) ) )
5554adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( ( A  /  B )  +  1 )  <  ( C  /  B )  <->  ( (
( A  /  B
)  +  1 )  x.  B )  < 
( ( C  /  B )  x.  B
) ) )
5651, 55mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( A  /  B
)  +  1 )  <  ( C  /  B ) )
575, 10, 14, 24, 56lelttrd 9616 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  < 
( C  /  B
) )
5857ex 434 1  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  <  ( C  -  B )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  < 
( C  /  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   CCcc 9367   RRcr 9368   0cc0 9369   1c1 9370    + caddc 9372    x. cmul 9374    < clt 9505    <_ cle 9506    - cmin 9682    / cdiv 10080   RR+crp 11078   |_cfl 11727
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-rp 11079  df-fl 11729
This theorem is referenced by: (None)
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