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Theorem modmulconst 11857
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 9297 . . . . 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 9319 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
433adant3 1019 . . . . 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 9297 . . . . . . 7 |- ( C e. NN -> C e. ZZ )
7 nnne0 8972 . . . . . . 7 |- ( C e. NN -> C =/= 0 )
86, 7jca 306 . . . . . 6 |- ( C e. NN -> ( C e. ZZ /\ C =/= 0 ) )
983ad2ant3 1022 . . . . 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 11854 . . . . 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 1249 . . 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 9283 . . . . . . . 8 |- ( A e. ZZ -> A e. CC )
15 zcn 9283 . . . . . . . 8 |- ( B e. ZZ -> B e. CC )
16 nncn 8952 . . . . . . . 8 |- ( C e. NN -> C e. CC )
1714, 15, 163anim123i 1186 . . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( A e. CC /\ B e. CC /\ C e. CC ) )
18 3anrot 985 . . . . . . 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 8367 . . . . . 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 4030 . . 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 999 . . . 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 1000 . . . 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 11833 . . 3 |- ( ( M e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
3125, 27, 29, 30syl3anc 1249 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
32 simpl3 1004 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> C e. NN )
3332, 25nnmulcld 8993 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. M ) e. NN )
3463ad2ant3 1022 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3534, 26zmulcld 9406 . . . 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 9406 . . . 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 11833 . . 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 1249 . 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 980   = wceq 1364   e. wcel 2160   =/= wne 2360   class class class wbr 4018  (class class class)co 5892   CCcc 7834   0cc0 7836   x. cmul 7841   - cmin 8153   NNcn 8944   ZZcz 9278   mod cmo 10348   || cdvds 11821
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-n0 9202  df-z 9279  df-q 9645  df-rp 9679  df-fl 10296  df-mod 10349  df-dvds 11822
This theorem is referenced by: (None)
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