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Theorem modmulnn 11192
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 9939 . . . . 5  |-  ( N  e.  NN  ->  N  e.  RR )
2 reflcl 11132 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
3 remulcl 9008 . . . . 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 9008 . . . . . 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 11132 . . . . 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 9955 . . . . . 6  |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  ( N  x.  M
)  e.  NN )
1211nnred 9947 . . . . 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 9940 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
15 nnne0 9964 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  =/=  0 )
1614, 15jca 519 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  e.  CC  /\  N  =/=  0 ) )
17 nncn 9940 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  CC )
18 nnne0 9964 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  =/=  0 )
1917, 18jca 519 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M  e.  CC  /\  M  =/=  0 ) )
20 mulne0 9596 . . . . . . . 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 9774 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) )  e.  RR )
24 reflcl 11132 . . . . 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 9049 . . 3  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  (
( N  x.  M
)  x.  ( |_
`  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) ) )  e.  RR )
27 nnnn0 10160 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
28 flmulnn0 11156 . . . . 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 9574 . 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 10579 . . . 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 11179 . . 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 11179 . . . 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 11168 . . . . . . 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 11168 . . . . . . . . 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 9047 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  CC )
45 divcan5 9648 . . . . . . . . . 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 5672 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  ( |_ `  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( |_ `  A
)  /  M ) ) )
48 recn 9013 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
49 divcan5 9648 . . . . . . . . . 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 5672 . . . . . . . 8  |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  ( ( N  x.  A )  / 
( N  x.  M
) ) )  =  ( |_ `  ( A  /  M ) ) )
5243, 47, 513eqtr4rd 2430 . . . . . . 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 2419 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( |_ `  ( ( |_
`  ( N  x.  A ) )  / 
( N  x.  M
) ) )  =  ( |_ `  (
( N  x.  ( |_ `  A ) )  /  ( N  x.  M ) ) ) )
5554oveq2d 6036 . . . 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 6036 . . 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 2419 . 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 4183 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 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923    x. cmul 8928    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   NN0cn0 10153   RR+crp 10544   |_cfl 11128    mod cmo 11177
This theorem is referenced by:  digit1  11440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fl 11129  df-mod 11178
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