Theorem modqaddmulmod 10391

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 1000	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2		qcn 9634	. . . . 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 1001	. . . . . . 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 )
7	4, 5, 6	modqcld 10328	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
8		simpl3 1002	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. ZZ )
9		zq 9626	. . . . . . 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 9637	. . . . . 6 |- ( ( ( B mod M ) e. QQ /\ C e. QQ ) -> ( ( B mod M ) x. C ) e. QQ )
12	7, 10, 11	syl2anc 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 9634	. . . . 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 8108	. . 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 5890	. 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 1020	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) -> C e. QQ )
18	17	adantr 276	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
19	7, 18, 11	syl2anc 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 9637	. . . . 5 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
21	4, 18, 20	syl2anc 411	. . . 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 10328	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) mod M ) e. QQ )
23		modqmulmod 10389	. . . . 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 1153	. . . 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 10358	. . . . 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 1238	. . . 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 2213	. . 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 10361	. 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 10363	. . . 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 1239	. . 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 9634	. . . . . 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 8108	. . . 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 5890	. . 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 2210	. 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 2214	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 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4004  (class class class)co 5875   CCcc 7809   0cc0 7811   + caddc 7814   x. cmul 7816   < clt 7992   ZZcz 9253   QQcq 9619   mod cmo 10322

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254  df-q 9620  df-rp 9654  df-fl 10270  df-mod 10323

This theorem is referenced by:  modprm0  12254  modprmn0modprm0  12256