Theorem modqaddmulmod 10195

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

Step	Hyp	Ref	Expression
1		simpl1 985	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2		qcn 9453	. . . . 5 |- ( A e. QQ -> A e. CC )
3	1, 2	syl 14	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. CC )
4		simpl2 986	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
5		simprl 521	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
6		simprr 522	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
7	4, 5, 6	modqcld 10132	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
8		simpl3 987	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. ZZ )
9		zq 9445	. . . . . . 7 |- ( C e. ZZ -> C e. QQ )
10	8, 9	syl 14	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
11		qmulcl 9456	. . . . . 6 |- ( ( ( B mod M ) e. QQ /\ C e. QQ ) -> ( ( B mod M ) x. C ) e. QQ )
12	7, 10, 11	syl2anc 409	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
13		qcn 9453	. . . . 5 |- ( ( ( B mod M ) x. C ) e. QQ -> ( ( B mod M ) x. C ) e. CC )
14	12, 13	syl 14	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. CC )
15	3, 14	addcomd 7937	. . 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 ) )
16	15	oveq1d 5797	. 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 ) )
17	9	3ad2ant3 1005	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) -> C e. QQ )
18	17	adantr 274	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
19	7, 18, 11	syl2anc 409	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
20		qmulcl 9456	. . . . 5 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
21	4, 18, 20	syl2anc 409	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. QQ )
22	21, 5, 6	modqcld 10132	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) mod M ) e. QQ )
23		modqmulmod 10193	. . . . 5 |- ( ( ( B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( B x. C ) mod M ) )
24	23	3adantl1 1138	. . . 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 10162	. . . . 5 |- ( ( ( B x. C ) e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( B x. C ) mod M ) mod M ) = ( ( B x. C ) mod M ) )
26	21, 5, 6, 25	syl3anc 1217	. . . 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 ) )
27	24, 26	eqtr4d 2176	. . 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 ) )
28	19, 22, 1, 5, 6, 27	modqadd1 10165	. 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 10167	. . . 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 ) )
30	21, 1, 5, 6, 29	syl22anc 1218	. . 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 9453	. . . . . 6 |- ( ( B x. C ) e. QQ -> ( B x. C ) e. CC )
32	21, 31	syl 14	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. CC )
33	32, 3	addcomd 7937	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) + A ) = ( A + ( B x. C ) ) )
34	33	oveq1d 5797	. . 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 ) )
35	30, 34	eqtrd 2173	. 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 ) )
36	16, 28, 35	3eqtrd 2177	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 ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   0cc0 7644   + caddc 7647   x. cmul 7649   < clt 7824   ZZcz 9078   QQcq 9438   mod cmo 10126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-q 9439  df-rp 9471  df-fl 10074  df-mod 10127

This theorem is referenced by: (None)