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Theorem modqcyc2 10333
Description: The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
Assertion
Ref Expression
modqcyc2  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  -  ( B  x.  N
) )  mod  B
)  =  ( A  mod  B ) )

Proof of Theorem modqcyc2
StepHypRef Expression
1 simplr 528 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  ->  N  e.  ZZ )
21zcnd 9352 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  ->  N  e.  CC )
3 qcn 9610 . . . . . . . 8  |-  ( B  e.  QQ  ->  B  e.  CC )
43ad2antrl 490 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  ->  B  e.  CC )
52, 4mulneg1d 8345 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( -u N  x.  B
)  =  -u ( N  x.  B )
)
6 mulcom 7918 . . . . . . . 8  |-  ( ( B  e.  CC  /\  N  e.  CC )  ->  ( B  x.  N
)  =  ( N  x.  B ) )
76negeqd 8129 . . . . . . 7  |-  ( ( B  e.  CC  /\  N  e.  CC )  -> 
-u ( B  x.  N )  =  -u ( N  x.  B
) )
84, 2, 7syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  ->  -u ( B  x.  N
)  =  -u ( N  x.  B )
)
95, 8eqtr4d 2213 . . . . 5  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( -u N  x.  B
)  =  -u ( B  x.  N )
)
109oveq2d 5884 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( A  +  (
-u N  x.  B
) )  =  ( A  +  -u ( B  x.  N )
) )
11 qcn 9610 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
1211ad2antrr 488 . . . . 5  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  ->  A  e.  CC )
134, 2mulcld 7955 . . . . 5  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( B  x.  N
)  e.  CC )
1412, 13negsubd 8251 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( A  +  -u ( B  x.  N
) )  =  ( A  -  ( B  x.  N ) ) )
1510, 14eqtr2d 2211 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( A  -  ( B  x.  N )
)  =  ( A  +  ( -u N  x.  B ) ) )
1615oveq1d 5883 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  -  ( B  x.  N
) )  mod  B
)  =  ( ( A  +  ( -u N  x.  B )
)  mod  B )
)
17 znegcl 9260 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
18 modqcyc 10332 . . 3  |-  ( ( ( A  e.  QQ  /\  -u N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  +  ( -u N  x.  B
) )  mod  B
)  =  ( A  mod  B ) )
1917, 18sylanl2 403 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  +  ( -u N  x.  B
) )  mod  B
)  =  ( A  mod  B ) )
2016, 19eqtrd 2210 1  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  -  ( B  x.  N
) )  mod  B
)  =  ( A  mod  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   class class class wbr 4000  (class class class)co 5868   CCcc 7787   0cc0 7789    + caddc 7792    x. cmul 7794    < clt 7969    - cmin 8105   -ucneg 8106   ZZcz 9229   QQcq 9595    mod cmo 10295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-n0 9153  df-z 9230  df-q 9596  df-rp 9628  df-fl 10243  df-mod 10296
This theorem is referenced by:  modqadd1  10334  modqmul1  10350  q2submod  10358  modqsubdir  10366
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