MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quoremnn0 Structured version   Unicode version

Theorem quoremnn0 11807
Description: Quotient and remainder of a nonnegative integer divided by a positive integer. (Contributed by NM, 14-Aug-2008.)
Hypotheses
Ref Expression
quorem.1  |-  Q  =  ( |_ `  ( A  /  B ) )
quorem.2  |-  R  =  ( A  -  ( B  x.  Q )
)
Assertion
Ref Expression
quoremnn0  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )

Proof of Theorem quoremnn0
StepHypRef Expression
1 quorem.1 . . 3  |-  Q  =  ( |_ `  ( A  /  B ) )
2 fldivnn0 11780 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  NN0 )
31, 2syl5eqel 2544 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  NN0 )
4 nn0z 10775 . . 3  |-  ( A  e.  NN0  ->  A  e.  ZZ )
5 quorem.2 . . . 4  |-  R  =  ( A  -  ( B  x.  Q )
)
61, 5quoremz 11806 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
74, 6sylan 471 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
8 simpl 457 . . . . . 6  |-  ( ( Q  e.  NN0  /\  Q  e.  ZZ )  ->  Q  e.  NN0 )
98anim1i 568 . . . . 5  |-  ( ( ( Q  e.  NN0  /\  Q  e.  ZZ )  /\  R  e.  NN0 )  ->  ( Q  e. 
NN0  /\  R  e.  NN0 ) )
109anasss 647 . . . 4  |-  ( ( Q  e.  NN0  /\  ( Q  e.  ZZ  /\  R  e.  NN0 )
)  ->  ( Q  e.  NN0  /\  R  e. 
NN0 ) )
1110anim1i 568 . . 3  |-  ( ( ( Q  e.  NN0  /\  ( Q  e.  ZZ  /\  R  e.  NN0 )
)  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R
) ) )  -> 
( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
1211anasss 647 . 2  |-  ( ( Q  e.  NN0  /\  ( ( Q  e.  ZZ  /\  R  e. 
NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )  ->  ( ( Q  e.  NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
133, 7, 12syl2anc 661 1  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195    + caddc 9391    x. cmul 9393    < clt 9524    - cmin 9701    / cdiv 10099   NNcn 10428   NN0cn0 10685   ZZcz 10752   |_cfl 11752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fl 11754
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator