ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intfracq Unicode version

Theorem intfracq 10276
Description: Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10275. (Contributed by NM, 16-Aug-2008.)
Hypotheses
Ref Expression
intfracq.1  |-  Z  =  ( |_ `  ( M  /  N ) )
intfracq.2  |-  F  =  ( ( M  /  N )  -  Z
)
Assertion
Ref Expression
intfracq  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( 0  <_  F  /\  F  <_  ( ( N  -  1 )  /  N )  /\  ( M  /  N
)  =  ( Z  +  F ) ) )

Proof of Theorem intfracq
StepHypRef Expression
1 znq 9583 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  QQ )
2 intfracq.1 . . . . 5  |-  Z  =  ( |_ `  ( M  /  N ) )
3 intfracq.2 . . . . 5  |-  F  =  ( ( M  /  N )  -  Z
)
42, 3intqfrac2 10275 . . . 4  |-  ( ( M  /  N )  e.  QQ  ->  (
0  <_  F  /\  F  <  1  /\  ( M  /  N )  =  ( Z  +  F
) ) )
51, 4syl 14 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( 0  <_  F  /\  F  <  1  /\  ( M  /  N
)  =  ( Z  +  F ) ) )
65simp1d 1004 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <_  F )
7 qfraclt1 10236 . . . . . . 7  |-  ( ( M  /  N )  e.  QQ  ->  (
( M  /  N
)  -  ( |_
`  ( M  /  N ) ) )  <  1 )
81, 7syl 14 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )  <  1 )
92oveq2i 5864 . . . . . . . 8  |-  ( ( M  /  N )  -  Z )  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
103, 9eqtri 2191 . . . . . . 7  |-  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
1110a1i 9 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N
) ) ) )
12 simpr 109 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  NN )
1312nncnd 8892 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
1412nnap0d 8924 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N #  0 )
1513, 14dividapd 8703 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  /  N
)  =  1 )
168, 11, 153brtr4d 4021 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <  ( N  /  N ) )
17 qre 9584 . . . . . . . . 9  |-  ( ( M  /  N )  e.  QQ  ->  ( M  /  N )  e.  RR )
181, 17syl 14 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  RR )
191flqcld 10233 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
202, 19eqeltrid 2257 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  ZZ )
2120zred 9334 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  RR )
2218, 21resubcld 8300 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  Z
)  e.  RR )
233, 22eqeltrid 2257 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  e.  RR )
24 nnre 8885 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR )
2524adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  RR )
26 nngt0 8903 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2724, 26jca 304 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  e.  RR  /\  0  <  N ) )
2827adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  e.  RR  /\  0  <  N ) )
29 ltmuldiv2 8791 . . . . . 6  |-  ( ( F  e.  RR  /\  N  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3023, 25, 28, 29syl3anc 1233 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3116, 30mpbird 166 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <  N )
323oveq2i 5864 . . . . . . 7  |-  ( N  x.  F )  =  ( N  x.  (
( M  /  N
)  -  Z ) )
3318recnd 7948 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
3420zcnd 9335 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  CC )
3513, 33, 34subdid 8333 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  (
( M  /  N
)  -  Z ) )  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
3632, 35eqtrid 2215 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
37 zcn 9217 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
3837adantr 274 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3938, 13, 14divcanap2d 8709 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
40 simpl 108 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
4139, 40eqeltrd 2247 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  e.  ZZ )
42 nnz 9231 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4342adantl 275 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
4443, 20zmulcld 9340 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  Z
)  e.  ZZ )
4541, 44zsubcld 9339 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  ( M  /  N
) )  -  ( N  x.  Z )
)  e.  ZZ )
4636, 45eqeltrd 2247 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  e.  ZZ )
47 zltlem1 9269 . . . . 5  |-  ( ( ( N  x.  F
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
4846, 43, 47syl2anc 409 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
4931, 48mpbid 146 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <_  ( N  -  1 ) )
50 peano2rem 8186 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
5124, 50syl 14 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  RR )
5251adantl 275 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  1 )  e.  RR )
53 lemuldiv2 8798 . . . 4  |-  ( ( F  e.  RR  /\  ( N  -  1
)  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  ->  ( ( N  x.  F )  <_  ( N  -  1 )  <->  F  <_  ( ( N  -  1 )  /  N ) ) )
5423, 52, 28, 53syl3anc 1233 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <_  ( N  -  1 )  <-> 
F  <_  ( ( N  -  1 )  /  N ) ) )
5549, 54mpbid 146 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <_  ( ( N  -  1 )  /  N ) )
565simp3d 1006 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  =  ( Z  +  F ) )
576, 55, 563jca 1172 1  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( 0  <_  F  /\  F  <_  ( ( N  -  1 )  /  N )  /\  ( M  /  N
)  =  ( Z  +  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    < clt 7954    <_ cle 7955    - cmin 8090    / cdiv 8589   NNcn 8878   ZZcz 9212   QQcq 9578   |_cfl 10224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226
This theorem is referenced by:  flqdiv  10277
  Copyright terms: Public domain W3C validator