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Theorem intfracq 11968
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 11967. (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 10864 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
21adantr 463 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  RR )
3 nnre 10538 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR )
43adantl 464 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  RR )
5 nnne0 10564 . . . . . 6  |-  ( N  e.  NN  ->  N  =/=  0 )
65adantl 464 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  =/=  0 )
72, 4, 6redivcld 10368 . . . 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 11967 . . . 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 1006 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <_  F )
13 fraclt1 11920 . . . . . . 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 6281 . . . . . . . 8  |-  ( ( M  /  N )  -  Z )  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
169, 15eqtri 2483 . . . . . . 7  |-  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N ) ) )
1716a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  =  ( ( M  /  N )  -  ( |_ `  ( M  /  N
) ) ) )
18 nncn 10539 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
1918, 5dividd 10314 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  /  N )  =  1 )
2019adantl 464 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  /  N
)  =  1 )
2114, 17, 203brtr4d 4469 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <  ( N  /  N ) )
22 reflcl 11914 . . . . . . . . . 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 2546 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  RR )
257, 24resubcld 9983 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  /  N )  -  Z
)  e.  RR )
269, 25syl5eqel 2546 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  e.  RR )
27 nngt0 10560 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
283, 27jca 530 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  e.  RR  /\  0  <  N ) )
2928adantl 464 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  e.  RR  /\  0  <  N ) )
30 ltmuldiv2 10412 . . . . . 6  |-  ( ( F  e.  RR  /\  N  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3126, 4, 29, 30syl3anc 1226 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  F  <  ( N  /  N ) ) )
3221, 31mpbird 232 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <  N )
339oveq2i 6281 . . . . . . 7  |-  ( N  x.  F )  =  ( N  x.  (
( M  /  N
)  -  Z ) )
3418adantl 464 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
357recnd 9611 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
367flcld 11916 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
378, 36syl5eqel 2546 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  ZZ )
3837zcnd 10966 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  Z  e.  CC )
3934, 35, 38subdid 10008 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  (
( M  /  N
)  -  Z ) )  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
4033, 39syl5eq 2507 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  =  ( ( N  x.  ( M  /  N ) )  -  ( N  x.  Z ) ) )
41 zcn 10865 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
4241adantr 463 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
4342, 34, 6divcan2d 10318 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
44 simpl 455 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
4543, 44eqeltrd 2542 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  e.  ZZ )
46 nnz 10882 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4746adantl 464 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
4847, 37zmulcld 10971 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  Z
)  e.  ZZ )
4945, 48zsubcld 10970 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  ( M  /  N
) )  -  ( N  x.  Z )
)  e.  ZZ )
5040, 49eqeltrd 2542 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  e.  ZZ )
51 zltlem1 10912 . . . . 5  |-  ( ( ( N  x.  F
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
5250, 47, 51syl2anc 659 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <  N  <->  ( N  x.  F )  <_  ( N  - 
1 ) ) )
5332, 52mpbid 210 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  F
)  <_  ( N  -  1 ) )
54 peano2rem 9877 . . . . . 6  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
553, 54syl 16 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  RR )
5655adantl 464 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  1 )  e.  RR )
57 lemuldiv2 10420 . . . 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 1226 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  x.  F )  <_  ( N  -  1 )  <-> 
F  <_  ( ( N  -  1 )  /  N ) ) )
5953, 58mpbid 210 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  F  <_  ( ( N  -  1 )  /  N ) )
6011simp3d 1008 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  =  ( Z  +  F ) )
6112, 59, 603jca 1174 1  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( 0  <_  F  /\  F  <_  ( ( N  -  1 )  /  N )  /\  ( M  /  N
)  =  ( Z  +  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   ZZcz 10860   |_cfl 11908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fl 11910
This theorem is referenced by:  fldiv  11969
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