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Theorem intfracq 11168
Description: Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 11167. (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 zre 10219 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
21adantr 452 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  RR )
3 nnre 9940 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR )
43adantl 453 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  RR )
5 nnne0 9965 . . . . . 6  |-  ( N  e.  NN  ->  N  =/=  0 )
65adantl 453 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  =/=  0 )
72, 4, 6redivcld 9775 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  RR )
8 intfracq.1 . . . . 5  |-  Z  =  ( |_ `  ( M  /  N ) )
9 intfracq.2 . . . . 5  |-  F  =  ( ( M  /  N )  -  Z
)
108, 9intfrac2 11167 . . . 4  |-  ( ( M  /  N )  e.  RR  ->  (
0  <_  F  /\  F  <  1  /\  ( M  /  N )  =  ( Z  +  F
) ) )
117, 10syl 16 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( 0  <_  F  /\  F  <  1  /\  ( M  /  N
)  =  ( Z  +  F ) ) )
1211simp1d 969 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <_  F )
13 fraclt1 11139 . . . . . . 7  |-  ( ( M  /  N )  e.  RR  ->  (
( M  /  N
)  -  ( |_
`  ( M  /  N ) ) )  <  1 )
147, 13syl 16 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )  <  1 )
158oveq2i 6032 . . . . . . . 8  |-  ( ( M  /  N )  -  Z )  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
169, 15eqtri 2408 . . . . . . 7  |-  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
1716a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N
) ) ) )
18 nncn 9941 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
1918, 5dividd 9721 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  /  N )  =  1 )
2019adantl 453 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  /  N
)  =  1 )
2114, 17, 203brtr4d 4184 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <  ( N  /  N ) )
22 reflcl 11133 . . . . . . . . . 10  |-  ( ( M  /  N )  e.  RR  ->  ( |_ `  ( M  /  N ) )  e.  RR )
237, 22syl 16 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  RR )
248, 23syl5eqel 2472 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  RR )
257, 24resubcld 9398 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  Z
)  e.  RR )
269, 25syl5eqel 2472 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  e.  RR )
27 nngt0 9962 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
283, 27jca 519 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  e.  RR  /\  0  <  N ) )
2928adantl 453 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  e.  RR  /\  0  <  N ) )
30 ltmuldiv2 9814 . . . . . 6  |-  ( ( F  e.  RR  /\  N  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3126, 4, 29, 30syl3anc 1184 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3221, 31mpbird 224 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <  N )
339oveq2i 6032 . . . . . . 7  |-  ( N  x.  F )  =  ( N  x.  (
( M  /  N
)  -  Z ) )
3418adantl 453 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
357recnd 9048 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
367flcld 11135 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
378, 36syl5eqel 2472 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  ZZ )
3837zcnd 10309 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  CC )
3934, 35, 38subdid 9422 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  (
( M  /  N
)  -  Z ) )  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
4033, 39syl5eq 2432 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
41 zcn 10220 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
4241adantr 452 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
4342, 34, 6divcan2d 9725 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
44 simpl 444 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
4543, 44eqeltrd 2462 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  e.  ZZ )
46 nnz 10236 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4746adantl 453 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
4847, 37zmulcld 10314 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  Z
)  e.  ZZ )
4945, 48zsubcld 10313 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  ( M  /  N
) )  -  ( N  x.  Z )
)  e.  ZZ )
5040, 49eqeltrd 2462 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  e.  ZZ )
51 zltlem1 10261 . . . . 5  |-  ( ( ( N  x.  F
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
5250, 47, 51syl2anc 643 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
5332, 52mpbid 202 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <_  ( N  -  1 ) )
54 peano2rem 9300 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
553, 54syl 16 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  RR )
5655adantl 453 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  1 )  e.  RR )
57 lemuldiv2 9823 . . . 4  |-  ( ( F  e.  RR  /\  ( N  -  1
)  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  ->  ( ( N  x.  F )  <_  ( N  -  1 )  <->  F  <_  ( ( N  -  1 )  /  N ) ) )
5826, 56, 29, 57syl3anc 1184 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <_  ( N  -  1 )  <-> 
F  <_  ( ( N  -  1 )  /  N ) ) )
5953, 58mpbid 202 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <_  ( ( N  -  1 )  /  N ) )
6011simp3d 971 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  =  ( Z  +  F ) )
6112, 59, 603jca 1134 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    <_ cle 9055    - cmin 9224    / cdiv 9610   NNcn 9933   ZZcz 10215   |_cfl 11129
This theorem is referenced by:  fldiv  11169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fl 11130
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