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Theorem modmulnn 11257
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 9999 . . . . 5  |-  ( N  e.  NN  ->  N  e.  RR )
2 reflcl 11197 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
3 remulcl 9067 . . . . 5  |-  ( ( N  e.  RR  /\  ( |_ `  A )  e.  RR )  -> 
( N  x.  ( |_ `  A ) )  e.  RR )
41, 2, 3syl2an 464 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  e.  RR )
543adant3 977 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  ( |_ `  A ) )  e.  RR )
6 remulcl 9067 . . . . . 6  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  x.  A
)  e.  RR )
71, 6sylan 458 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  A
)  e.  RR )
8 reflcl 11197 . . . . 5  |-  ( ( N  x.  A )  e.  RR  ->  ( |_ `  ( N  x.  A ) )  e.  RR )
97, 8syl 16 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( |_ `  ( N  x.  A )
)  e.  RR )
1093adant3 977 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( N  x.  A ) )  e.  RR )
11 nnmulcl 10015 . . . . . 6  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  NN )
1211nnred 10007 . . . . 5  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  RR )
13123adant2 976 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  RR )
14 nncn 10000 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
15 nnne0 10024 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  =/=  0 )
1614, 15jca 519 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  e.  CC  /\  N  =/=  0 ) )
17 nncn 10000 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  CC )
18 nnne0 10024 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  =/=  0 )
1917, 18jca 519 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M  e.  CC  /\  M  =/=  0 ) )
20 mulne0 9656 . . . . . . . 8  |-  ( ( ( N  e.  CC  /\  N  =/=  0 )  /\  ( M  e.  CC  /\  M  =/=  0 ) )  -> 
( N  x.  M
)  =/=  0 )
2116, 19, 20syl2an 464 . . . . . . 7  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  =/=  0 )
22213adant2 976 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  =/=  0 )
235, 13, 22redivcld 9834 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  e.  RR )
24 reflcl 11197 . . . . 5  |-  ( ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  e.  RR  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) )  e.  RR )
2523, 24syl 16 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) )  e.  RR )
2613, 25remulcld 9108 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  M
)  x.  ( |_
`  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) ) )  e.  RR )
27 nnnn0 10220 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
28 flmulnn0 11221 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  <_  ( |_ `  ( N  x.  A
) ) )
2927, 28sylan 458 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) )  <_  ( |_ `  ( N  x.  A
) ) )
30293adant3 977 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  ( |_ `  A ) )  <_ 
( |_ `  ( N  x.  A )
) )
315, 10, 26, 30lesub1dd 9634 . 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 10639 . . . 4  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  RR+ )
33323adant2 976 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  RR+ )
34 modval 11244 . . 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 643 . 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 11244 . . . 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 643 . . 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 977 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  A )  e.  RR )
39113adant2 976 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( N  x.  M )  e.  NN )
40 fldiv 11233 . . . . . . 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 643 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  A
)  /  ( N  x.  M ) ) ) )
42 fldiv 11233 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  (
( |_ `  A
)  /  M ) )  =  ( |_
`  ( A  /  M ) ) )
43423adant3 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( |_
`  A )  /  M ) )  =  ( |_ `  ( A  /  M ) ) )
442recnd 9106 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  CC )
45 divcan5 9708 . . . . . . . . . 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 1226 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  =  ( ( |_ `  A )  /  M
) )
4746fveq2d 5724 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( |_ `  A
)  /  M ) ) )
48 recn 9072 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
49 divcan5 9708 . . . . . . . . . 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 1226 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( N  x.  A
)  /  ( N  x.  M ) )  =  ( A  /  M ) )
5150fveq2d 5724 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  ( A  /  M ) ) )
5243, 47, 513eqtr4rd 2478 . . . . . . 7  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
53523comr 1161 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
5441, 53eqtrd 2467 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
5554oveq2d 6089 . . . 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 6089 . . 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 2467 . 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 4234 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   RR+crp 10604   |_cfl 11193    mod cmo 11242
This theorem is referenced by:  digit1  11505
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
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