Theorem modqmul1 9448

Description: Multiplication property of the modulo operation. Note that the multiplier C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)

Hypotheses

Ref	Expression
modqmul1.a	|- ( ph -> A e. QQ )
modqmul1.b	|- ( ph -> B e. QQ )
modqmul1.c	|- ( ph -> C e. ZZ )
modqmul1.d	|- ( ph -> D e. QQ )
modqmul1.dgt0	|- ( ph -> 0 < D )
modqmul1.ab	|- ( ph -> ( A mod D ) = ( B mod D ) )

Assertion

Ref	Expression
modqmul1	|- ( ph -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) )

Proof of Theorem modqmul1

Step	Hyp	Ref	Expression
1		modqmul1.ab	. 2 |- ( ph -> ( A mod D ) = ( B mod D ) )
2		modqmul1.a	. . . . . . 7 |- ( ph -> A e. QQ )
3		modqmul1.d	. . . . . . 7 |- ( ph -> D e. QQ )
4		modqmul1.dgt0	. . . . . . 7 |- ( ph -> 0 < D )
5		modqval 9395	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1170	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqmul1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 9395	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1170	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2096	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 5544	. . . . 5 |- ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
12	10, 11	syl6bi 161	. . . 4 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) ) )
13		qcn 8789	. . . . . . . . . 10 |- ( D e. QQ -> D e. CC )
14	3, 13	syl 14	. . . . . . . . 9 |- ( ph -> D e. CC )
15		modqmul1.c	. . . . . . . . . 10 |- ( ph -> C e. ZZ )
16	15	zcnd 8540	. . . . . . . . 9 |- ( ph -> C e. CC )
17	4	gt0ne0d 7669	. . . . . . . . . . . 12 |- ( ph -> D =/= 0 )
18		qdivcl 8798	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
19	2, 3, 17, 18	syl3anc 1170	. . . . . . . . . . 11 |- ( ph -> ( A / D ) e. QQ )
20	19	flqcld 9348	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
21	20	zcnd 8540	. . . . . . . . 9 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
22	14, 16, 21	mulassd 7193	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) )
23	14, 16, 21	mul32d 7317	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
24	22, 23	eqtr3d 2116	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
25	24	oveq2d 5553	. . . . . 6 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
26		qcn 8789	. . . . . . . 8 |- ( A e. QQ -> A e. CC )
27	2, 26	syl 14	. . . . . . 7 |- ( ph -> A e. CC )
28	14, 21	mulcld 7190	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
29	27, 28, 16	subdird 7575	. . . . . 6 |- ( ph -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
30	25, 29	eqtr4d 2117	. . . . 5 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) )
31		qdivcl 8798	. . . . . . . . . . . 12 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
32	7, 3, 17, 31	syl3anc 1170	. . . . . . . . . . 11 |- ( ph -> ( B / D ) e. QQ )
33	32	flqcld 9348	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
34	33	zcnd 8540	. . . . . . . . 9 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
35	14, 16, 34	mulassd 7193	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) )
36	14, 16, 34	mul32d 7317	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
37	35, 36	eqtr3d 2116	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
38	37	oveq2d 5553	. . . . . 6 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
39		qcn 8789	. . . . . . . 8 |- ( B e. QQ -> B e. CC )
40	7, 39	syl 14	. . . . . . 7 |- ( ph -> B e. CC )
41	14, 34	mulcld 7190	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
42	40, 41, 16	subdird 7575	. . . . . 6 |- ( ph -> ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
43	38, 42	eqtr4d 2117	. . . . 5 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
44	30, 43	eqeq12d 2096	. . . 4 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) <-> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) ) )
45	12, 44	sylibrd 167	. . 3 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) ) )
46		oveq1 5544	. . . 4 |- ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) )
47		zq 8781	. . . . . . . 8 |- ( C e. ZZ -> C e. QQ )
48	15, 47	syl 14	. . . . . . 7 |- ( ph -> C e. QQ )
49		qmulcl 8792	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
50	2, 48, 49	syl2anc 403	. . . . . 6 |- ( ph -> ( A x. C ) e. QQ )
51	15, 20	zmulcld 8545	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( A / D ) ) ) e. ZZ )
52		modqcyc2 9431	. . . . . 6 |- ( ( ( ( A x. C ) e. QQ /\ ( C x. ( |_ ` ( A / D ) ) ) e. ZZ ) /\ ( D e. QQ /\ 0 < D ) ) -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( A x. C ) mod D ) )
53	50, 51, 3, 4, 52	syl22anc 1171	. . . . 5 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( A x. C ) mod D ) )
54		qmulcl 8792	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
55	7, 48, 54	syl2anc 403	. . . . . 6 |- ( ph -> ( B x. C ) e. QQ )
56	15, 33	zmulcld 8545	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( B / D ) ) ) e. ZZ )
57		modqcyc2 9431	. . . . . 6 |- ( ( ( ( B x. C ) e. QQ /\ ( C x. ( |_ ` ( B / D ) ) ) e. ZZ ) /\ ( D e. QQ /\ 0 < D ) ) -> ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) = ( ( B x. C ) mod D ) )
58	55, 56, 3, 4, 57	syl22anc 1171	. . . . 5 |- ( ph -> ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) = ( ( B x. C ) mod D ) )
59	53, 58	eqeq12d 2096	. . . 4 |- ( ph -> ( ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) <-> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) ) )
60	46, 59	syl5ib 152	. . 3 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) ) )
61	45, 60	syld 44	. 2 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) ) )
62	1, 61	mpd 13	1 |- ( ph -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   =/= wne 2246   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   CCcc 7030   0cc0 7032   x. cmul 7037   < clt 7204   - cmin 7335   / cdiv 7816   ZZcz 8421   QQcq 8774   |_cfl 9339   mod cmo 9393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145  ax-arch 7146

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-n0 8345  df-z 8422  df-q 8775  df-rp 8805  df-fl 9341  df-mod 9394

This theorem is referenced by:  modqmul12d  9449  modqnegd  9450  modqmulmod  9460