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Theorem modqaddmulmod 10752
Description: The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
Assertion
Ref Expression
modqaddmulmod |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Proof of Theorem modqaddmulmod
StepHypRef Expression
1 simpl1 1027 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2 qcn 9965 . . . . 5 |- ( A e. QQ -> A e. CC )
31, 2syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. CC )
4 simpl2 1028 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
5 simprl 531 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
6 simprr 533 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
74, 5, 6modqcld 10689 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
8 simpl3 1029 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. ZZ )
9 zq 9957 . . . . . . 7 |- ( C e. ZZ -> C e. QQ )
108, 9syl 14 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
11 qmulcl 9968 . . . . . 6 |- ( ( ( B mod M ) e. QQ /\ C e. QQ ) -> ( ( B mod M ) x. C ) e. QQ )
127, 10, 11syl2anc 411 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
13 qcn 9965 . . . . 5 |- ( ( ( B mod M ) x. C ) e. QQ -> ( ( B mod M ) x. C ) e. CC )
1412, 13syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. CC )
153, 14addcomd 8423 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A + ( ( B mod M ) x. C ) ) = ( ( ( B mod M ) x. C ) + A ) )
1615oveq1d 6064 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( ( ( B mod M ) x. C ) + A ) mod M ) )
1793ad2ant3 1047 . . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) -> C e. QQ )
1817adantr 276 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
197, 18, 11syl2anc 411 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
20 qmulcl 9968 . . . . 5 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
214, 18, 20syl2anc 411 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. QQ )
2221, 5, 6modqcld 10689 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) mod M ) e. QQ )
23 modqmulmod 10750 . . . . 5 |- ( ( ( B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( B x. C ) mod M ) )
24233adantl1 1180 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( B x. C ) mod M ) )
25 modqabs2 10719 . . . . 5 |- ( ( ( B x. C ) e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( B x. C ) mod M ) mod M ) = ( ( B x. C ) mod M ) )
2621, 5, 6, 25syl3anc 1274 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B x. C ) mod M ) mod M ) = ( ( B x. C ) mod M ) )
2724, 26eqtr4d 2268 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( ( B x. C ) mod M ) mod M ) )
2819, 22, 1, 5, 6, 27modqadd1 10722 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B mod M ) x. C ) + A ) mod M ) = ( ( ( ( B x. C ) mod M ) + A ) mod M ) )
29 modqaddmod 10724 . . . 4 |- ( ( ( ( B x. C ) e. QQ /\ A e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( ( B x. C ) + A ) mod M ) )
3021, 1, 5, 6, 29syl22anc 1275 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( ( B x. C ) + A ) mod M ) )
31 qcn 9965 . . . . . 6 |- ( ( B x. C ) e. QQ -> ( B x. C ) e. CC )
3221, 31syl 14 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. CC )
3332, 3addcomd 8423 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) + A ) = ( A + ( B x. C ) ) )
3433oveq1d 6064 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B x. C ) + A ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
3530, 34eqtrd 2265 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
3616, 28, 353eqtrd 2269 1 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8124   0cc0 8126   + caddc 8129   x. cmul 8131   < clt 8307   ZZcz 9576   QQcq 9950   mod cmo 10683
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-n0 9496  df-z 9577  df-q 9951  df-rp 9986  df-fl 10629  df-mod 10684
This theorem is referenced by:  modprm0  12948  modprmn0modprm0  12950
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