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

Theorem modmulnn 10988
Description: Move a natural number in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)
Assertion
Ref Expression
modmulnn  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) )  <_ 
( ( |_ `  ( N  x.  A
) )  mod  ( N  x.  M )
) )

Proof of Theorem modmulnn
StepHypRef Expression
1 nnre 9753 . . . . 5  |-  ( N  e.  NN  ->  N  e.  RR )
2 reflcl 10928 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
3 remulcl 8822 . . . . 5  |-  ( ( N  e.  RR  /\  ( |_ `  A )  e.  RR )  -> 
( N  x.  ( |_ `  A ) )  e.  RR )
41, 2, 3syl2an 463 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  e.  RR )
543adant3 975 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  ( |_ `  A ) )  e.  RR )
6 remulcl 8822 . . . . . 6  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  x.  A
)  e.  RR )
71, 6sylan 457 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  A
)  e.  RR )
8 reflcl 10928 . . . . 5  |-  ( ( N  x.  A )  e.  RR  ->  ( |_ `  ( N  x.  A ) )  e.  RR )
97, 8syl 15 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( |_ `  ( N  x.  A )
)  e.  RR )
1093adant3 975 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( N  x.  A ) )  e.  RR )
11 nnmulcl 9769 . . . . . 6  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  NN )
1211nnred 9761 . . . . 5  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  RR )
13123adant2 974 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  RR )
14 nncn 9754 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
15 nnne0 9778 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  =/=  0 )
1614, 15jca 518 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  e.  CC  /\  N  =/=  0 ) )
17 nncn 9754 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  CC )
18 nnne0 9778 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  =/=  0 )
1917, 18jca 518 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M  e.  CC  /\  M  =/=  0 ) )
20 mulne0 9410 . . . . . . . 8  |-  ( ( ( N  e.  CC  /\  N  =/=  0 )  /\  ( M  e.  CC  /\  M  =/=  0 ) )  -> 
( N  x.  M
)  =/=  0 )
2116, 19, 20syl2an 463 . . . . . . 7  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  =/=  0 )
22213adant2 974 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  =/=  0 )
235, 13, 22redivcld 9588 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  e.  RR )
24 reflcl 10928 . . . . 5  |-  ( ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  e.  RR  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) )  e.  RR )
2523, 24syl 15 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) )  e.  RR )
2613, 25remulcld 8863 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  M
)  x.  ( |_
`  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) ) )  e.  RR )
27 nnnn0 9972 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
28 flmulnn0 10952 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  <_  ( |_ `  ( N  x.  A
) ) )
2927, 28sylan 457 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  <_  ( |_ `  ( N  x.  A
) ) )
30293adant3 975 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  ( |_ `  A ) )  <_ 
( |_ `  ( N  x.  A )
) )
315, 10, 26, 30lesub1dd 9388 . 2  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  -  ( ( N  x.  M )  x.  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) ) )  <_  (
( |_ `  ( N  x.  A )
)  -  ( ( N  x.  M )  x.  ( |_ `  ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) ) ) )
3211nnrpd 10389 . . . 4  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  RR+ )
33323adant2 974 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  RR+ )
34 modval 10975 . . 3  |-  ( ( ( N  x.  ( |_ `  A ) )  e.  RR  /\  ( N  x.  M )  e.  RR+ )  ->  (
( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) )  =  ( ( N  x.  ( |_ `  A ) )  -  ( ( N  x.  M )  x.  ( |_ `  ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) ) ) )
355, 33, 34syl2anc 642 . 2  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) )  =  ( ( N  x.  ( |_ `  A ) )  -  ( ( N  x.  M )  x.  ( |_ `  ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) ) ) )
36 modval 10975 . . . 4  |-  ( ( ( |_ `  ( N  x.  A )
)  e.  RR  /\  ( N  x.  M
)  e.  RR+ )  ->  ( ( |_ `  ( N  x.  A
) )  mod  ( N  x.  M )
)  =  ( ( |_ `  ( N  x.  A ) )  -  ( ( N  x.  M )  x.  ( |_ `  (
( |_ `  ( N  x.  A )
)  /  ( N  x.  M ) ) ) ) ) )
3710, 33, 36syl2anc 642 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( |_ `  ( N  x.  A )
)  mod  ( N  x.  M ) )  =  ( ( |_ `  ( N  x.  A
) )  -  (
( N  x.  M
)  x.  ( |_
`  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) ) ) ) )
3873adant3 975 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  A )  e.  RR )
39113adant2 974 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  NN )
40 fldiv 10964 . . . . . . 7  |-  ( ( ( N  x.  A
)  e.  RR  /\  ( N  x.  M
)  e.  NN )  ->  ( |_ `  ( ( |_ `  ( N  x.  A
) )  /  ( N  x.  M )
) )  =  ( |_ `  ( ( N  x.  A )  /  ( N  x.  M ) ) ) )
4138, 39, 40syl2anc 642 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  A
)  /  ( N  x.  M ) ) ) )
42 fldiv 10964 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  (
( |_ `  A
)  /  M ) )  =  ( |_
`  ( A  /  M ) ) )
43423adant3 975 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( |_
`  A )  /  M ) )  =  ( |_ `  ( A  /  M ) ) )
442recnd 8861 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  CC )
45 divcan5 9462 . . . . . . . . . 10  |-  ( ( ( |_ `  A
)  e.  CC  /\  ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  =  ( ( |_
`  A )  /  M ) )
4644, 19, 16, 45syl3an 1224 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  =  ( ( |_ `  A )  /  M
) )
4746fveq2d 5529 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( |_ `  A
)  /  M ) ) )
48 recn 8827 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
49 divcan5 9462 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  e.  CC  /\  M  =/=  0 )  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( ( N  x.  A )  /  ( N  x.  M )
)  =  ( A  /  M ) )
5048, 19, 16, 49syl3an 1224 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( N  x.  A
)  /  ( N  x.  M ) )  =  ( A  /  M ) )
5150fveq2d 5529 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  ( A  /  M ) ) )
5243, 47, 513eqtr4rd 2326 . . . . . . 7  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
53523comr 1159 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
5441, 53eqtrd 2315 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
5554oveq2d 5874 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  M
)  x.  ( |_
`  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) ) )  =  ( ( N  x.  M )  x.  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) ) )
5655oveq2d 5874 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( |_ `  ( N  x.  A )
)  -  ( ( N  x.  M )  x.  ( |_ `  ( ( |_ `  ( N  x.  A
) )  /  ( N  x.  M )
) ) ) )  =  ( ( |_
`  ( N  x.  A ) )  -  ( ( N  x.  M )  x.  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) ) ) ) )
5737, 56eqtrd 2315 . 2  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( |_ `  ( N  x.  A )
)  mod  ( N  x.  M ) )  =  ( ( |_ `  ( N  x.  A
) )  -  (
( N  x.  M
)  x.  ( |_
`  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) ) ) ) )
5831, 35, 573brtr4d 4053 1  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) )  <_ 
( ( |_ `  ( N  x.  A
) )  mod  ( N  x.  M )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   RR+crp 10354   |_cfl 10924    mod cmo 10973
This theorem is referenced by:  digit1  11235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
  Copyright terms: Public domain W3C validator