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Theorem quoremz 10959
Description: Quotient and remainder of an integer divided by a natural number. TO DO - is this really needed for anything? Should we use  mod to simplify it? (Contributed by NM, 14-Aug-2008.)
Hypotheses
Ref Expression
quorem.1  |-  Q  =  ( |_ `  ( A  /  B ) )
quorem.2  |-  R  =  ( A  -  ( B  x.  Q )
)
Assertion
Ref Expression
quoremz  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )

Proof of Theorem quoremz
StepHypRef Expression
1 quorem.1 . . 3  |-  Q  =  ( |_ `  ( A  /  B ) )
2 zre 10028 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
32adantr 451 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  RR )
4 nnre 9753 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  RR )
54adantl 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  RR )
6 nnne0 9778 . . . . . 6  |-  ( B  e.  NN  ->  B  =/=  0 )
76adantl 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
83, 5, 7redivcld 9588 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
98flcld 10930 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
101, 9syl5eqel 2367 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  e.  ZZ )
11 quorem.2 . . 3  |-  R  =  ( A  -  ( B  x.  Q )
)
1210zcnd 10118 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  e.  CC )
13 nncn 9754 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
1413adantl 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
1512, 14, 7divcan3d 9541 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
16 flle 10931 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
178, 16syl 15 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
181, 17syl5eqbr 4056 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
1915, 18eqbrtrd 4043 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
20 nnz 10045 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
2120adantl 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
2221, 10zmulcld 10123 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  ZZ )
2322zred 10117 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  RR )
24 nngt0 9775 . . . . . . 7  |-  ( B  e.  NN  ->  0  <  B )
2524adantl 452 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  0  <  B )
26 lediv1 9621 . . . . . 6  |-  ( ( ( B  x.  Q
)  e.  RR  /\  A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2723, 3, 5, 25, 26syl112anc 1186 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2819, 27mpbird 223 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  <_  A )
29 simpl 443 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
30 znn0sub 10065 . . . . 5  |-  ( ( ( B  x.  Q
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( B  x.  Q )  <_  A  <->  ( A  -  ( B  x.  Q ) )  e.  NN0 ) )
3122, 29, 30syl2anc 642 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( A  -  ( B  x.  Q ) )  e.  NN0 ) )
3228, 31mpbid 201 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
3311, 32syl5eqel 2367 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  e.  NN0 )
341oveq2i 5869 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
35 fraclt1 10934 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
368, 35syl 15 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
3734, 36syl5eqbr 4056 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
3811oveq1i 5868 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
39 zcn 10029 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  CC )
4039adantr 451 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
4122zcnd 10118 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  CC )
4213, 6jca 518 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
4342adantl 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
44 divsubdir 9456 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( B  x.  Q
)  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  -  ( B  x.  Q ) )  /  B )  =  ( ( A  /  B
)  -  ( ( B  x.  Q )  /  B ) ) )
4540, 41, 43, 44syl3anc 1182 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
4615oveq2d 5874 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
4745, 46eqtrd 2315 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
4838, 47syl5eq 2327 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
4913, 6dividd 9534 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
5049adantl 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
5137, 48, 503brtr4d 4053 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  /  B
)  <  ( B  /  B ) )
5233nn0red 10019 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  e.  RR )
53 ltdiv1 9620 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5452, 5, 5, 25, 53syl112anc 1186 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5551, 54mpbird 223 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  <  B )
5611oveq2i 5869 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
5741, 40pncan3d 9160 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
5856, 57syl5req 2328 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
5955, 58jca 518 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
6010, 33, 59jca31 520 1  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   |_cfl 10924
This theorem is referenced by:  quoremnn0  10960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fl 10925
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