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Theorem modqcyc2 9442
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 497 . . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. ZZ )
21zcnd 8551 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. CC )
3 qcn 8800 . . . . . . . 8 |- ( B e. QQ -> B e. CC )
43ad2antrl 474 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> B e. CC )
52, 4mulneg1d 7582 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( N x. B ) )
6 mulcom 7164 . . . . . . . 8 |- ( ( B e. CC /\ N e. CC ) -> ( B x. N ) = ( N x. B ) )
76negeqd 7370 . . . . . . 7 |- ( ( B e. CC /\ N e. CC ) -> -u ( B x. N ) = -u ( N x. B ) )
84, 2, 7syl2anc 403 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> -u ( B x. N ) = -u ( N x. B ) )
95, 8eqtr4d 2117 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( B x. N ) )
109oveq2d 5559 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + ( -u N x. B ) ) = ( A + -u ( B x. N ) ) )
11 qcn 8800 . . . . . 6 |- ( A e. QQ -> A e. CC )
1211ad2antrr 472 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> A e. CC )
134, 2mulcld 7201 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( B x. N ) e. CC )
1412, 13negsubd 7492 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + -u ( B x. N ) ) = ( A - ( B x. N ) ) )
1510, 14eqtr2d 2115 . . 3 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A - ( B x. N ) ) = ( A + ( -u N x. B ) ) )
1615oveq1d 5558 . 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 8463 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
18 modqcyc 9441 . . 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 395 . 2 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A + ( -u N x. B ) ) mod B ) = ( A mod B ) )
2016, 19eqtrd 2114 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 102   = wceq 1285   e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   0cc0 7043   + caddc 7046   x. cmul 7048   < clt 7215   - cmin 7346   -ucneg 7347   ZZcz 8432   QQcq 8785   mod cmo 9404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-q 8786  df-rp 8816  df-fl 9352  df-mod 9405
This theorem is referenced by:  modqadd1  9443  modqmul1  9459  q2submod  9467  modqsubdir  9475
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