ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  modqmulmodr Unicode version

Theorem modqmulmodr 10333
Description: The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
Assertion
Ref Expression
modqmulmodr  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( A  x.  B )  mod 
M ) )

Proof of Theorem modqmulmodr
StepHypRef Expression
1 simpll 524 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  A  e.  ZZ )
21zcnd 9322 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  A  e.  CC )
3 simplr 525 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  B  e.  QQ )
4 simprl 526 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  M  e.  QQ )
5 simprr 527 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <  M )
63, 4, 5modqcld 10271 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( B  mod  M
)  e.  QQ )
7 qcn 9580 . . . . 5  |-  ( ( B  mod  M )  e.  QQ  ->  ( B  mod  M )  e.  CC )
86, 7syl 14 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( B  mod  M
)  e.  CC )
92, 8mulcomd 7928 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  x.  ( B  mod  M ) )  =  ( ( B  mod  M )  x.  A ) )
109oveq1d 5865 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( ( B  mod  M )  x.  A )  mod 
M ) )
11 modqmulmod 10332 . . 3  |-  ( ( ( B  e.  QQ  /\  A  e.  ZZ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( B  mod  M )  x.  A )  mod  M
)  =  ( ( B  x.  A )  mod  M ) )
1211ancom1s 564 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( B  mod  M )  x.  A )  mod  M
)  =  ( ( B  x.  A )  mod  M ) )
13 qcn 9580 . . . . 5  |-  ( B  e.  QQ  ->  B  e.  CC )
143, 13syl 14 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  B  e.  CC )
1514, 2mulcomd 7928 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( B  x.  A
)  =  ( A  x.  B ) )
1615oveq1d 5865 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( B  x.  A )  mod  M
)  =  ( ( A  x.  B )  mod  M ) )
1710, 12, 163eqtrd 2207 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( A  x.  B )  mod 
M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761    x. cmul 7766    < clt 7941   ZZcz 9199   QQcq 9565    mod cmo 10265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-n0 9123  df-z 9200  df-q 9566  df-rp 9598  df-fl 10213  df-mod 10266
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator