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Theorem modcyc 12011
Description: The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)
Assertion
Ref Expression
modcyc  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  N  e.  ZZ )  ->  (
( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )

Proof of Theorem modcyc
StepHypRef Expression
1 zre 10880 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 rpre 11238 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  e.  RR )
3 remulcl 9589 . . . . . . . 8  |-  ( ( N  e.  RR  /\  B  e.  RR )  ->  ( N  x.  B
)  e.  RR )
41, 2, 3syl2an 477 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  B  e.  RR+ )  -> 
( N  x.  B
)  e.  RR )
5 readdcl 9587 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( N  x.  B
)  e.  RR )  ->  ( A  +  ( N  x.  B
) )  e.  RR )
64, 5sylan2 474 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  e.  ZZ  /\  B  e.  RR+ )
)  ->  ( A  +  ( N  x.  B ) )  e.  RR )
763impb 1192 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( A  +  ( N  x.  B ) )  e.  RR )
8 simp3 998 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  B  e.  RR+ )
9 modval 11978 . . . . 5  |-  ( ( ( A  +  ( N  x.  B ) )  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  +  ( N  x.  B
) )  mod  B
)  =  ( ( A  +  ( N  x.  B ) )  -  ( B  x.  ( |_ `  ( ( A  +  ( N  x.  B ) )  /  B ) ) ) ) )
107, 8, 9syl2anc 661 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  mod  B )  =  ( ( A  +  ( N  x.  B ) )  -  ( B  x.  ( |_ `  ( ( A  +  ( N  x.  B ) )  /  B ) ) ) ) )
11 recn 9594 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
12113ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  A  e.  CC )
134recnd 9634 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  B  e.  RR+ )  -> 
( N  x.  B
)  e.  CC )
14133adant1 1014 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( N  x.  B )  e.  CC )
15 rpcnne0 11249 . . . . . . . . . . . 12  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
16153ad2ant3 1019 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
17 divdir 10242 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( N  x.  B
)  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  +  ( N  x.  B ) )  /  B )  =  ( ( A  /  B
)  +  ( ( N  x.  B )  /  B ) ) )
1812, 14, 16, 17syl3anc 1228 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  /  B )  =  ( ( A  /  B )  +  ( ( N  x.  B )  /  B
) ) )
19 zcn 10881 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
20 divcan4 10244 . . . . . . . . . . . . . 14  |-  ( ( N  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( N  x.  B
)  /  B )  =  N )
21203expb 1197 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( N  x.  B )  /  B )  =  N )
2219, 15, 21syl2an 477 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  B  e.  RR+ )  -> 
( ( N  x.  B )  /  B
)  =  N )
23223adant1 1014 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( N  x.  B
)  /  B )  =  N )
2423oveq2d 6311 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  /  B
)  +  ( ( N  x.  B )  /  B ) )  =  ( ( A  /  B )  +  N ) )
2518, 24eqtrd 2508 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  /  B )  =  ( ( A  /  B )  +  N ) )
2625fveq2d 5876 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( |_ `  ( ( A  +  ( N  x.  B ) )  /  B ) )  =  ( |_ `  (
( A  /  B
)  +  N ) ) )
27 rerpdivcl 11259 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
28273adant2 1015 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR )
29 simp2 997 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  N  e.  ZZ )
30 fladdz 11938 . . . . . . . . 9  |-  ( ( ( A  /  B
)  e.  RR  /\  N  e.  ZZ )  ->  ( |_ `  (
( A  /  B
)  +  N ) )  =  ( ( |_ `  ( A  /  B ) )  +  N ) )
3128, 29, 30syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( |_ `  ( ( A  /  B )  +  N ) )  =  ( ( |_ `  ( A  /  B
) )  +  N
) )
3226, 31eqtrd 2508 . . . . . . 7  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( |_ `  ( ( A  +  ( N  x.  B ) )  /  B ) )  =  ( ( |_ `  ( A  /  B
) )  +  N
) )
3332oveq2d 6311 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  x.  ( |_ `  ( ( A  +  ( N  x.  B
) )  /  B
) ) )  =  ( B  x.  (
( |_ `  ( A  /  B ) )  +  N ) ) )
34 rpcn 11240 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  e.  CC )
35343ad2ant3 1019 . . . . . . 7  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  B  e.  CC )
36 reflcl 11913 . . . . . . . . . 10  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
3736recnd 9634 . . . . . . . . 9  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  CC )
3827, 37syl 16 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
39383adant2 1015 . . . . . . 7  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( |_ `  ( A  /  B ) )  e.  CC )
40193ad2ant2 1018 . . . . . . 7  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  N  e.  CC )
4135, 39, 40adddid 9632 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  x.  ( ( |_ `  ( A  /  B ) )  +  N ) )  =  ( ( B  x.  ( |_ `  ( A  /  B ) ) )  +  ( B  x.  N ) ) )
42 mulcom 9590 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  B  e.  CC )  ->  ( N  x.  B
)  =  ( B  x.  N ) )
4319, 34, 42syl2an 477 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  B  e.  RR+ )  -> 
( N  x.  B
)  =  ( B  x.  N ) )
44433adant1 1014 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( N  x.  B )  =  ( B  x.  N ) )
4544eqcomd 2475 . . . . . . 7  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  x.  N )  =  ( N  x.  B ) )
4645oveq2d 6311 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( B  x.  ( |_ `  ( A  /  B ) ) )  +  ( B  x.  N ) )  =  ( ( B  x.  ( |_ `  ( A  /  B ) ) )  +  ( N  x.  B ) ) )
4733, 41, 463eqtrd 2512 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  x.  ( |_ `  ( ( A  +  ( N  x.  B
) )  /  B
) ) )  =  ( ( B  x.  ( |_ `  ( A  /  B ) ) )  +  ( N  x.  B ) ) )
4847oveq2d 6311 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  -  ( B  x.  ( |_ `  ( ( A  +  ( N  x.  B
) )  /  B
) ) ) )  =  ( ( A  +  ( N  x.  B ) )  -  ( ( B  x.  ( |_ `  ( A  /  B ) ) )  +  ( N  x.  B ) ) ) )
4934adantl 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
5049, 38mulcld 9628 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
51503adant2 1015 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( B  x.  ( |_ `  ( A  /  B
) ) )  e.  CC )
5212, 51, 14pnpcan2d 9980 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  -  ( ( B  x.  ( |_
`  ( A  /  B ) ) )  +  ( N  x.  B ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
5310, 48, 523eqtrd 2512 . . 3  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
54 modval 11978 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
55543adant2 1015 . . 3  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
5653, 55eqtr4d 2511 . 2  |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  B  e.  RR+ )  ->  (
( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )
57563com23 1202 1  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  N  e.  ZZ )  ->  (
( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    + caddc 9507    x. cmul 9509    - cmin 9817    / cdiv 10218   ZZcz 10876   RR+crp 11232   |_cfl 11907    mod cmo 11976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-mod 11977
This theorem is referenced by:  modcyc2  12012  modxai  14430  wilthlem1  23208  wilthlem2  23209  lgsdir2lem1  23464  lgsdir2lem5  23468  lgseisenlem1  23490  dirkerper  31719  sqwvfoura  31852  sqwvfourb  31853  fourierswlem  31854  fouriersw  31855
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