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Theorem modqcyc2 10020
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 502 . . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. ZZ )
21zcnd 9072 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. CC )
3 qcn 9322 . . . . . . . 8 |- ( B e. QQ -> B e. CC )
43ad2antrl 479 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> B e. CC )
52, 4mulneg1d 8086 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( N x. B ) )
6 mulcom 7667 . . . . . . . 8 |- ( ( B e. CC /\ N e. CC ) -> ( B x. N ) = ( N x. B ) )
76negeqd 7874 . . . . . . 7 |- ( ( B e. CC /\ N e. CC ) -> -u ( B x. N ) = -u ( N x. B ) )
84, 2, 7syl2anc 406 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> -u ( B x. N ) = -u ( N x. B ) )
95, 8eqtr4d 2148 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( B x. N ) )
109oveq2d 5742 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + ( -u N x. B ) ) = ( A + -u ( B x. N ) ) )
11 qcn 9322 . . . . . 6 |- ( A e. QQ -> A e. CC )
1211ad2antrr 477 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> A e. CC )
134, 2mulcld 7704 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( B x. N ) e. CC )
1412, 13negsubd 7996 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + -u ( B x. N ) ) = ( A - ( B x. N ) ) )
1510, 14eqtr2d 2146 . . 3 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A - ( B x. N ) ) = ( A + ( -u N x. B ) ) )
1615oveq1d 5741 . 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 8983 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
18 modqcyc 10019 . . 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 398 . 2 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A + ( -u N x. B ) ) mod B ) = ( A mod B ) )
2016, 19eqtrd 2145 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 103   = wceq 1312   e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   0cc0 7541   + caddc 7544   x. cmul 7546   < clt 7718   - cmin 7850   -ucneg 7851   ZZcz 8952   QQcq 9307   mod cmo 9982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-n0 8876  df-z 8953  df-q 9308  df-rp 9338  df-fl 9930  df-mod 9983
This theorem is referenced by:  modqadd1  10021  modqmul1  10037  q2submod  10045  modqsubdir  10053
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