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Theorem ltdifltdiv 12016
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 12007 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
2 peano2re 9757 . . . . . 6  |-  ( ( |_ `  ( A  /  B ) )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
31, 2syl 17 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( |_ `  ( A  /  B
) )  +  1 )  e.  RR )
433adant3 1025 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
54adantr 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  e.  RR )
6 rerpdivcl 11281 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
7 peano2re 9757 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  +  1 )  e.  RR )
86, 7syl 17 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  +  1 )  e.  RR )
983adant3 1025 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  +  1 )  e.  RR )
109adantr 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( A  /  B
)  +  1 )  e.  RR )
11 rerpdivcl 11281 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR+ )  -> 
( C  /  B
)  e.  RR )
1211ancoms 454 . . . . 5  |-  ( ( B  e.  RR+  /\  C  e.  RR )  ->  ( C  /  B )  e.  RR )
13123adant1 1023 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( C  /  B )  e.  RR )
1413adantr 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( C  /  B )  e.  RR )
1513adant3 1025 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
1615adantr 466 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
1763adant3 1025 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
1817adantr 466 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  /  B )  e.  RR )
19 1red 9609 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  1  e.  RR )
20 3simpa 1002 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
2120adantr 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
22 fldivle 12013 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
2321, 22syl 17 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
2416, 18, 19, 23leadd1dd 10178 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  <_ 
( ( A  /  B )  +  1 ) )
25 rpre 11259 . . . . . . 7  |-  ( B  e.  RR+  ->  B  e.  RR )
26 ltaddsub 10039 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
2725, 26syl3an2 1298 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
2827biimpar 487 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  ( A  +  B )  <  C )
29 recn 9580 . . . . . . . . . . 11  |-  ( ( A  /  B )  e.  RR  ->  ( A  /  B )  e.  CC )
306, 29syl 17 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
31303adant3 1025 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  /  B )  e.  CC )
32 1cnd 9610 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  1  e.  CC )
33 rpcn 11261 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  B  e.  CC )
34333ad2ant2 1027 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  CC )
3531, 32, 34adddird 9619 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( ( ( A  /  B )  x.  B )  +  ( 1  x.  B
) ) )
36 recn 9580 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
37363ad2ant1 1026 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  A  e.  CC )
38 rpne0 11268 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  B  =/=  0 )
39383ad2ant2 1027 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  =/=  0 )
4037, 34, 39divcan1d 10335 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  x.  B )  =  A )
4133mulid2d 9612 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  ( 1  x.  B )  =  B )
42413ad2ant2 1027 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
1  x.  B )  =  B )
4340, 42oveq12d 6267 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  x.  B
)  +  ( 1  x.  B ) )  =  ( A  +  B ) )
4435, 43eqtrd 2462 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( A  +  B ) )
45 recn 9580 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
46453ad2ant3 1028 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  C  e.  CC )
4746, 34, 39divcan1d 10335 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( C  /  B
)  x.  B )  =  C )
4844, 47breq12d 4379 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( ( A  /  B )  +  1 )  x.  B
)  <  ( ( C  /  B )  x.  B )  <->  ( A  +  B )  <  C
) )
4948adantr 466 . . . . 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
) )
5028, 49mpbird 235 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( ( A  /  B )  +  1 )  x.  B )  <  ( ( C  /  B )  x.  B ) )
5117, 7syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  +  1 )  e.  RR )
52 simp2 1006 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  RR+ )
5351, 13, 52ltmul1d 11330 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  <  ( C  /  B )  <->  ( (
( A  /  B
)  +  1 )  x.  B )  < 
( ( C  /  B )  x.  B
) ) )
5453adantr 466 . . . 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
) ) )
5550, 54mpbird 235 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( A  /  B
)  +  1 )  <  ( C  /  B ) )
565, 10, 14, 24, 55lelttrd 9744 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  < 
( C  /  B
) )
5756ex 435 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    < clt 9626    <_ cle 9627    - cmin 9811    / cdiv 10220   RR+crp 11253   |_cfl 11976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fl 11978
This theorem is referenced by: (None)
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