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Theorem modqaddmulmod 10500
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 1002 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2 qcn 9725 . . . . 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 1003 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
5 simprl 529 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
6 simprr 531 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
74, 5, 6modqcld 10437 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
8 simpl3 1004 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. ZZ )
9 zq 9717 . . . . . . 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 9728 . . . . . 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 9725 . . . . 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 8194 . . 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 5940 . 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 1022 . . . . 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 9728 . . . . 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 10437 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) mod M ) e. QQ )
23 modqmulmod 10498 . . . . 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 1155 . . . 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 10467 . . . . 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 1249 . . . 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 2232 . . 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 10470 . 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 10472 . . . 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 1250 . . 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 9725 . . . . . 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 8194 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) + A ) = ( A + ( B x. C ) ) )
3433oveq1d 5940 . . 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 2229 . 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 2233 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899   x. cmul 7901   < clt 8078   ZZcz 9343   QQcq 9710   mod cmo 10431
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-q 9711  df-rp 9746  df-fl 10377  df-mod 10432
This theorem is referenced by:  modprm0  12448  modprmn0modprm0  12450
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