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Theorem quoremz 12082
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 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  RR )
4 nnre 10616 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  RR )
54adantl 468 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  RR )
6 nnne0 10642 . . . . . 6  |-  ( B  e.  NN  ->  B  =/=  0 )
76adantl 468 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
83, 5, 7redivcld 10435 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
98flcld 12034 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
101, 9syl5eqel 2533 . 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 468 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
1512, 14, 7divcan3d 10388 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
16 flle 12035 . . . . . . . 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 4436 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
1915, 18eqbrtrd 4423 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
20 nnz 10959 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
2120adantl 468 . . . . . . . 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 468 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  0  <  B )
26 lediv1 10470 . . . . . 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 1272 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2819, 27mpbird 236 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  x.  Q
)  <_  A )
29 simpl 459 . . . . 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 667 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( A  -  ( B  x.  Q ) )  e.  NN0 ) )
3228, 31mpbid 214 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
3311, 32syl5eqel 2533 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  e.  NN0 )
341oveq2i 6301 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
35 fraclt1 12038 . . . . . . 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 4436 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
3811oveq1i 6300 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
39 zcn 10942 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  CC )
4039adantr 467 . . . . . . . 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 535 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
4342adantl 468 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
44 divsubdir 10303 . . . . . . . 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 1268 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
4615oveq2d 6306 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
4745, 46eqtrd 2485 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
4838, 47syl5eq 2497 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
4913, 6dividd 10381 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
5049adantl 468 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
5137, 48, 503brtr4d 4433 . . . 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 10469 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5452, 5, 5, 25, 53syl112anc 1272 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5551, 54mpbird 236 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  R  <  B )
5611oveq2i 6301 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
5741, 40pncan3d 9989 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
5856, 57syl5req 2498 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
5955, 58jca 535 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
6010, 33, 59jca31 537 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 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   |_cfl 12026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fl 12028
This theorem is referenced by:  quoremnn0  12083
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