Theorem modqmul1 10451

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 10398	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1249	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqmul1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10398	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1249	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2208	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 5926	. . . . 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	biimtrdi 163	. . . 4 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) ) )
13		qcn 9702	. . . . . . . . . 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 9443	. . . . . . . . 9 |- ( ph -> C e. CC )
17	4	gt0ne0d 8533	. . . . . . . . . . . 12 |- ( ph -> D =/= 0 )
18		qdivcl 9711	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
19	2, 3, 17, 18	syl3anc 1249	. . . . . . . . . . 11 |- ( ph -> ( A / D ) e. QQ )
20	19	flqcld 10349	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
21	20	zcnd 9443	. . . . . . . . 9 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
22	14, 16, 21	mulassd 8045	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) )
23	14, 16, 21	mul32d 8174	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
24	22, 23	eqtr3d 2228	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
25	24	oveq2d 5935	. . . . . 6 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
26		qcn 9702	. . . . . . . 8 |- ( A e. QQ -> A e. CC )
27	2, 26	syl 14	. . . . . . 7 |- ( ph -> A e. CC )
28	14, 21	mulcld 8042	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
29	27, 28, 16	subdird 8436	. . . . . 6 |- ( ph -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
30	25, 29	eqtr4d 2229	. . . . 5 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) )
31		qdivcl 9711	. . . . . . . . . . . 12 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
32	7, 3, 17, 31	syl3anc 1249	. . . . . . . . . . 11 |- ( ph -> ( B / D ) e. QQ )
33	32	flqcld 10349	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
34	33	zcnd 9443	. . . . . . . . 9 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
35	14, 16, 34	mulassd 8045	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) )
36	14, 16, 34	mul32d 8174	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
37	35, 36	eqtr3d 2228	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
38	37	oveq2d 5935	. . . . . 6 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
39		qcn 9702	. . . . . . . 8 |- ( B e. QQ -> B e. CC )
40	7, 39	syl 14	. . . . . . 7 |- ( ph -> B e. CC )
41	14, 34	mulcld 8042	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
42	40, 41, 16	subdird 8436	. . . . . 6 |- ( ph -> ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
43	38, 42	eqtr4d 2229	. . . . 5 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
44	30, 43	eqeq12d 2208	. . . 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 169	. . 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 5926	. . . 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 9694	. . . . . . . 8 |- ( C e. ZZ -> C e. QQ )
48	15, 47	syl 14	. . . . . . 7 |- ( ph -> C e. QQ )
49		qmulcl 9705	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
50	2, 48, 49	syl2anc 411	. . . . . 6 |- ( ph -> ( A x. C ) e. QQ )
51	15, 20	zmulcld 9448	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( A / D ) ) ) e. ZZ )
52		modqcyc2 10434	. . . . . 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 1250	. . . . 5 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( A x. C ) mod D ) )
54		qmulcl 9705	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
55	7, 48, 54	syl2anc 411	. . . . . 6 |- ( ph -> ( B x. C ) e. QQ )
56	15, 33	zmulcld 9448	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( B / D ) ) ) e. ZZ )
57		modqcyc2 10434	. . . . . 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 1250	. . . . 5 |- ( ph -> ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) = ( ( B x. C ) mod D ) )
59	53, 58	eqeq12d 2208	. . . 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	imbitrid 154	. . 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 45	. 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 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   x. cmul 7879   < clt 8056   - cmin 8192   / cdiv 8693   ZZcz 9320   QQcq 9687   |_cfl 10340   mod cmo 10396

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-q 9688  df-rp 9723  df-fl 10342  df-mod 10397

This theorem is referenced by:  modqmul12d  10452  modqnegd  10453  modqmulmod  10463  eulerthlema  12371  fermltl  12375  odzdvds  12386  lgsdir2lem4  15188  lgsdirprm  15191  gausslemma2d  15226