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Theorem modmulconst 12615
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 9674 . . . . 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 9696 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
433adant3 1048 . . . . 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 9674 . . . . . . 7 |- ( C e. NN -> C e. ZZ )
7 nnne0 9341 . . . . . . 7 |- ( C e. NN -> C =/= 0 )
86, 7jca 306 . . . . . 6 |- ( C e. NN -> ( C e. ZZ /\ C =/= 0 ) )
983ad2ant3 1051 . . . . 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 12612 . . . . 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 1278 . . 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 9660 . . . . . . . 8 |- ( A e. ZZ -> A e. CC )
15 zcn 9660 . . . . . . . 8 |- ( B e. ZZ -> B e. CC )
16 nncn 9321 . . . . . . . 8 |- ( C e. NN -> C e. CC )
1714, 15, 163anim123i 1215 . . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( A e. CC /\ B e. CC /\ C e. CC ) )
18 3anrot 1014 . . . . . . 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 8719 . . . . . 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 4143 . . 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 1028 . . . 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 1029 . . . 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 12591 . . 3 |- ( ( M e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
3125, 27, 29, 30syl3anc 1278 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
32 simpl3 1033 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> C e. NN )
3332, 25nnmulcld 9362 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( C x. M ) e. NN )
3463ad2ant3 1051 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3534, 26zmulcld 9785 . . . 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 9785 . . . 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 12591 . . 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 1278 . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4131  (class class class)co 6088   CCcc 8183   0cc0 8185   x. cmul 8190   - cmin 8504   NNcn 9313   ZZcz 9655   mod cmo 10780   || cdvds 12579
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-dvds 12580
This theorem is used by: (None)
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