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Theorem ltdifltdiv 11969
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 11960 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
2 peano2re 9770 . . . . . 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 1016 . . . 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 11272 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
7 peano2re 9770 . . . . . 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 1016 . . . 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 11272 . . . . . 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 1014 . . . 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 1016 . . . . 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 1016 . . . . 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 9628 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  1  e.  RR )
20 3simpa 993 . . . . . 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 11966 . . . . 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 10187 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  <_ 
( ( A  /  B )  +  1 ) )
25 rpre 11251 . . . . . . 7  |-  ( B  e.  RR+  ->  B  e.  RR )
26 ltaddsub 10047 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
2725, 26syl3an2 1262 . . . . . 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 9599 . . . . . . . . . . 11  |-  ( ( A  /  B )  e.  RR  ->  ( A  /  B )  e.  CC )
306, 29syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
31303adant3 1016 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  ( A  /  B )  e.  CC )
32 1cnd 9629 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  1  e.  CC )
33 rpcn 11253 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  B  e.  CC )
34333ad2ant2 1018 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  CC )
3531, 32, 34adddird 9638 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( ( ( A  /  B )  x.  B )  +  ( 1  x.  B
) ) )
36 recn 9599 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
37363ad2ant1 1017 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  A  e.  CC )
38 rpne0 11260 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  B  =/=  0 )
39383ad2ant2 1018 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  =/=  0 )
4037, 34, 39divcan1d 10342 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  x.  B )  =  A )
4133mulid2d 9631 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  ( 1  x.  B )  =  B )
42413ad2ant2 1018 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
1  x.  B )  =  B )
4340, 42oveq12d 6314 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  x.  B
)  +  ( 1  x.  B ) )  =  ( A  +  B ) )
4435, 43eqtrd 2498 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( A  /  B )  +  1 )  x.  B )  =  ( A  +  B ) )
45 recn 9599 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
46453ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  C  e.  CC )
4746, 34, 39divcan1d 10342 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( C  /  B
)  x.  B )  =  C )
4844, 47breq12d 4469 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( ( ( A  /  B )  +  1 )  x.  B
)  <  ( ( C  /  B )  x.  B )  <->  ( A  +  B )  <  C
) )
4948adantr 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
) )
5028, 49mpbird 232 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( ( A  /  B )  +  1 )  x.  B )  <  ( ( C  /  B )  x.  B ) )
5117, 7syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  (
( A  /  B
)  +  1 )  e.  RR )
52 simp2 997 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  ->  B  e.  RR+ )
5351, 13, 52ltmul1d 11318 . . . . 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 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
) ) )
5550, 54mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( A  /  B
)  +  1 )  <  ( C  /  B ) )
565, 10, 14, 24, 55lelttrd 9757 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR )  /\  A  <  ( C  -  B
) )  ->  (
( |_ `  ( A  /  B ) )  +  1 )  < 
( C  /  B
) )
5756ex 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   RR+crp 11245   |_cfl 11930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fl 11932
This theorem is referenced by: (None)
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