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Theorem quoremz 12079
Description: Quotient and remainder of an integer divided by a positive integer. 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 10941 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
32adantr 466 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  RR )
4 nnre 10616 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  RR )
54adantl 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  RR )
6 nnne0 10642 . . . . . 6  |-  ( B  e.  NN  ->  B  =/=  0 )
76adantl 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
83, 5, 7redivcld 10434 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
98flcld 12031 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
101, 9syl5eqel 2521 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  e.  ZZ )
11 quorem.2 . . 3  |-  R  =  ( A  -  ( B  x.  Q )
)
1210zcnd 11041 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  e.  CC )
13 nncn 10617 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
1413adantl 467 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
1512, 14, 7divcan3d 10387 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
16 flle 12032 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
178, 16syl 17 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
181, 17syl5eqbr 4459 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
1915, 18eqbrtrd 4446 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
20 nnz 10959 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
2120adantl 467 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
2221, 10zmulcld 11046 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  ZZ )
2322zred 11040 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  RR )
24 nngt0 10638 . . . . . . 7  |-  ( B  e.  NN  ->  0  <  B )
2524adantl 467 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  0  <  B )
26 lediv1 10469 . . . . . 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 1268 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2819, 27mpbird 235 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  <_  A )
29 simpl 458 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
30 znn0sub 10984 . . . . 5  |-  ( ( ( B  x.  Q
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( B  x.  Q )  <_  A  <->  ( A  -  ( B  x.  Q ) )  e.  NN0 ) )
3122, 29, 30syl2anc 665 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( A  -  ( B  x.  Q ) )  e.  NN0 ) )
3228, 31mpbid 213 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
3311, 32syl5eqel 2521 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  e.  NN0 )
341oveq2i 6316 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
35 fraclt1 12035 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
368, 35syl 17 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
3734, 36syl5eqbr 4459 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
3811oveq1i 6315 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
39 zcn 10942 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  CC )
4039adantr 466 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
4122zcnd 11041 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  CC )
4213, 6jca 534 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
4342adantl 467 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
44 divsubdir 10302 . . . . . . . 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 1264 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
4615oveq2d 6321 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
4745, 46eqtrd 2470 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
4838, 47syl5eq 2482 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
4913, 6dividd 10380 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
5049adantl 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
5137, 48, 503brtr4d 4456 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  /  B
)  <  ( B  /  B ) )
5233nn0red 10926 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  e.  RR )
53 ltdiv1 10468 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5452, 5, 5, 25, 53syl112anc 1268 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5551, 54mpbird 235 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  <  B )
5611oveq2i 6316 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
5741, 40pncan3d 9988 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
5856, 57syl5req 2483 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
5955, 58jca 534 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
6010, 33, 59jca31 536 1  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   NN0cn0 10869   ZZcz 10937   |_cfl 12023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fl 12025
This theorem is referenced by:  quoremnn0  12080
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