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

Theorem moddiffl 11251
Description: The modulo operation differs from  A by an integer multiple of  B. (Contributed by Mario Carneiro, 6-Sep-2016.)
Assertion
Ref Expression
moddiffl  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B
) ) )

Proof of Theorem moddiffl
StepHypRef Expression
1 modval 11244 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
21oveq2d 6089 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  ( A  mod  B ) )  =  ( A  -  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) ) )
3 simpl 444 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  RR )
43recnd 9106 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
5 rpcn 10612 . . . . . . 7  |-  ( B  e.  RR+  ->  B  e.  CC )
65adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
7 rerpdivcl 10631 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
87flcld 11199 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  ZZ )
98zcnd 10368 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
106, 9mulcld 9100 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
114, 10nncand 9408 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) )
122, 11eqtrd 2467 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  ( A  mod  B ) )  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) )
1312oveq1d 6088 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( A  mod  B ) )  /  B )  =  ( ( B  x.  ( |_ `  ( A  /  B
) ) )  /  B ) )
14 rpne0 10619 . . . 4  |-  ( B  e.  RR+  ->  B  =/=  0 )
1514adantl 453 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  =/=  0 )
169, 6, 15divcan3d 9787 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( |_ `  ( A  /  B ) ) )  /  B )  =  ( |_ `  ( A  /  B
) ) )
1713, 16eqtrd 2467 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    - cmin 9283    / cdiv 9669   RR+crp 10604   |_cfl 11193    mod cmo 11242
This theorem is referenced by:  moddifz  11252  bitsinv1lem  12945  bitsres  12977  rdr  26434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243
  Copyright terms: Public domain W3C validator