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Theorem modmulconst 12320
Description: Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
Assertion
Ref Expression
modmulconst |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> ( ( C x. A ) mod ( C x. M ) ) = ( ( C x. B ) mod ( C x. M ) ) ) )
Proof of Theorem modmulconst
StepHypRef Expression
1 nnz 9453 . . . . 5 |- ( M e. NN -> M e. ZZ )
21adantl 277 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> M e. ZZ )
3 zsubcl 9475 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
433adant3 1041 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( A - B ) e. ZZ )
54adantr 276 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( A - B ) e. ZZ )
6 nnz 9453 . . . . . . 7 |- ( C e. NN -> C e. ZZ )
7 nnne0 9126 . . . . . . 7 |- ( C e. NN -> C =/= 0 )
86, 7jca 306 . . . . . 6 |- ( C e. NN -> ( C e. ZZ /\ C =/= 0 ) )
983ad2ant3 1044 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C e. ZZ /\ C =/= 0 ) )
109adantr 276 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C e. ZZ /\ C =/= 0 ) )
11 dvdscmulr 12317 . . . . 5 |- ( ( M e. ZZ /\ ( A - B ) e. ZZ /\ ( C e. ZZ /\ C =/= 0 ) ) -> ( ( C x. M ) || ( C x. ( A - B ) ) <-> M || ( A - B ) ) )
1211bicomd 141 . . . 4 |- ( ( M e. ZZ /\ ( A - B ) e. ZZ /\ ( C e. ZZ /\ C =/= 0 ) ) -> ( M || ( A - B ) <-> ( C x. M ) || ( C x. ( A - B ) ) ) )
132, 5, 10, 12syl3anc 1271 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( M || ( A - B ) <-> ( C x. M ) || ( C x. ( A - B ) ) ) )
14 zcn 9439 . . . . . . . 8 |- ( A e. ZZ -> A e. CC )
15 zcn 9439 . . . . . . . 8 |- ( B e. ZZ -> B e. CC )
16 nncn 9106 . . . . . . . 8 |- ( C e. NN -> C e. CC )
1714, 15, 163anim123i 1208 . . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( A e. CC /\ B e. CC /\ C e. CC ) )
18 3anrot 1007 . . . . . . 7 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) <-> ( A e. CC /\ B e. CC /\ C e. CC ) )
1917, 18sylibr 134 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C e. CC /\ A e. CC /\ B e. CC ) )
20 subdi 8519 . . . . . 6 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
2119, 20syl 14 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
2221adantr 276 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
2322breq2d 4094 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( C x. M ) || ( C x. ( A - B ) ) <-> ( C x. M ) || ( ( C x. A ) - ( C x. B ) ) ) )
2413, 23bitrd 188 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( M || ( A - B ) <-> ( C x. M ) || ( ( C x. A ) - ( C x. B ) ) ) )
25 simpr 110 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> M e. NN )
26 simp1 1021 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> A e. ZZ )
2726adantr 276 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> A e. ZZ )
28 simp2 1022 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> B e. ZZ )
2928adantr 276 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> B e. ZZ )
30 moddvds 12296 . . 3 |- ( ( M e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
3125, 27, 29, 30syl3anc 1271 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
32 simpl3 1026 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> C e. NN )
3332, 25nnmulcld 9147 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. M ) e. NN )
3463ad2ant3 1044 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3534, 26zmulcld 9563 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C x. A ) e. ZZ )
3635adantr 276 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. A ) e. ZZ )
3734, 28zmulcld 9563 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C x. B ) e. ZZ )
3837adantr 276 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. B ) e. ZZ )
39 moddvds 12296 . . 3 |- ( ( ( C x. M ) e. NN /\ ( C x. A ) e. ZZ /\ ( C x. B ) e. ZZ ) -> ( ( ( C x. A ) mod ( C x. M ) ) = ( ( C x. B ) mod ( C x. M ) ) <-> ( C x. M ) || ( ( C x. A ) - ( C x. B ) ) ) )
4033, 36, 38, 39syl3anc 1271 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( ( C x. A ) mod ( C x. M ) ) = ( ( C x. B ) mod ( C x. M ) ) <-> ( C x. M ) || ( ( C x. A ) - ( C x. B ) ) ) )
4124, 31, 403bitr4d 220 1 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> ( ( C x. A ) mod ( C x. M ) ) = ( ( C x. B ) mod ( C x. M ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082  (class class class)co 5994   CCcc 7985   0cc0 7987   x. cmul 7992   - cmin 8305   NNcn 9098   ZZcz 9434   mod cmo 10531   || cdvds 12284
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-po 4384  df-iso 4385  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-n0 9358  df-z 9435  df-q 9803  df-rp 9838  df-fl 10477  df-mod 10532  df-dvds 12285
This theorem is referenced by: (None)
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