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Theorem intfracq 10219
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 10218. (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 9533 . . . 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 10218 . . . 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 994 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <_  F )
7 qfraclt1 10179 . . . . . . 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 5835 . . . . . . . 8  |-  ( ( M  /  N )  -  Z )  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
103, 9eqtri 2178 . . . . . . 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 8847 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
1412nnap0d 8879 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N #  0 )
1513, 14dividapd 8659 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  /  N
)  =  1 )
168, 11, 153brtr4d 3996 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <  ( N  /  N ) )
17 qre 9534 . . . . . . . . 9  |-  ( ( M  /  N )  e.  QQ  ->  ( M  /  N )  e.  RR )
181, 17syl 14 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  RR )
191flqcld 10176 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
202, 19eqeltrid 2244 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  ZZ )
2120zred 9286 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  RR )
2218, 21resubcld 8256 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  Z
)  e.  RR )
233, 22eqeltrid 2244 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  e.  RR )
24 nnre 8840 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR )
2524adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  RR )
26 nngt0 8858 . . . . . . . 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 8746 . . . . . 6  |-  ( ( F  e.  RR  /\  N  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3023, 25, 28, 29syl3anc 1220 . . . . 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 5835 . . . . . . 7  |-  ( N  x.  F )  =  ( N  x.  (
( M  /  N
)  -  Z ) )
3318recnd 7906 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
3420zcnd 9287 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  CC )
3513, 33, 34subdid 8289 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  (
( M  /  N
)  -  Z ) )  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
3632, 35syl5eq 2202 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
37 zcn 9172 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
3837adantr 274 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3938, 13, 14divcanap2d 8665 . . . . . . . 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 2234 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  e.  ZZ )
42 nnz 9186 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4342adantl 275 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
4443, 20zmulcld 9292 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  Z
)  e.  ZZ )
4541, 44zsubcld 9291 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  ( M  /  N
) )  -  ( N  x.  Z )
)  e.  ZZ )
4636, 45eqeltrd 2234 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  e.  ZZ )
47 zltlem1 9224 . . . . 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 8142 . . . . . 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 8753 . . . 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 1220 . . 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 996 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  =  ( Z  +  F ) )
576, 55, 563jca 1162 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 963    = wceq 1335    e. wcel 2128   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731   0cc0 7732   1c1 7733    + caddc 7735    x. cmul 7737    < clt 7912    <_ cle 7913    - cmin 8046    / cdiv 8545   NNcn 8833   ZZcz 9167   QQcq 9528   |_cfl 10167
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-q 9529  df-rp 9561  df-fl 10169
This theorem is referenced by:  flqdiv  10220
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