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Theorem modqcyc2 10316
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 525 . . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. ZZ )
21zcnd 9335 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> N e. CC )
3 qcn 9593 . . . . . . . 8 |- ( B e. QQ -> B e. CC )
43ad2antrl 487 . . . . . . 7 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> B e. CC )
52, 4mulneg1d 8330 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( N x. B ) )
6 mulcom 7903 . . . . . . . 8 |- ( ( B e. CC /\ N e. CC ) -> ( B x. N ) = ( N x. B ) )
76negeqd 8114 . . . . . . 7 |- ( ( B e. CC /\ N e. CC ) -> -u ( B x. N ) = -u ( N x. B ) )
84, 2, 7syl2anc 409 . . . . . 6 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> -u ( B x. N ) = -u ( N x. B ) )
95, 8eqtr4d 2206 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( -u N x. B ) = -u ( B x. N ) )
109oveq2d 5869 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + ( -u N x. B ) ) = ( A + -u ( B x. N ) ) )
11 qcn 9593 . . . . . 6 |- ( A e. QQ -> A e. CC )
1211ad2antrr 485 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> A e. CC )
134, 2mulcld 7940 . . . . 5 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( B x. N ) e. CC )
1412, 13negsubd 8236 . . . 4 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A + -u ( B x. N ) ) = ( A - ( B x. N ) ) )
1510, 14eqtr2d 2204 . . 3 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( A - ( B x. N ) ) = ( A + ( -u N x. B ) ) )
1615oveq1d 5868 . 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 9243 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
18 modqcyc 10315 . . 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 401 . 2 |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A + ( -u N x. B ) ) mod B ) = ( A mod B ) )
2016, 19eqtrd 2203 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 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   + caddc 7777   x. cmul 7779   < clt 7954   - cmin 8090   -ucneg 8091   ZZcz 9212   QQcq 9578   mod cmo 10278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226  df-mod 10279
This theorem is referenced by:  modqadd1  10317  modqmul1  10333  q2submod  10341  modqsubdir  10349
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