Theorem modqmul1 10486

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 10433	. . . . . . 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 10433	. . . . . . 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 2211	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 5932	. . . . 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 9725	. . . . . . . . . 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 9466	. . . . . . . . 9 |- ( ph -> C e. CC )
17	4	gt0ne0d 8556	. . . . . . . . . . . 12 |- ( ph -> D =/= 0 )
18		qdivcl 9734	. . . . . . . . . . . 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 10384	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
21	20	zcnd 9466	. . . . . . . . 9 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
22	14, 16, 21	mulassd 8067	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) )
23	14, 16, 21	mul32d 8196	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
24	22, 23	eqtr3d 2231	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
25	24	oveq2d 5941	. . . . . 6 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
26		qcn 9725	. . . . . . . 8 |- ( A e. QQ -> A e. CC )
27	2, 26	syl 14	. . . . . . 7 |- ( ph -> A e. CC )
28	14, 21	mulcld 8064	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
29	27, 28, 16	subdird 8458	. . . . . 6 |- ( ph -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
30	25, 29	eqtr4d 2232	. . . . 5 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) )
31		qdivcl 9734	. . . . . . . . . . . 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 10384	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
34	33	zcnd 9466	. . . . . . . . 9 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
35	14, 16, 34	mulassd 8067	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) )
36	14, 16, 34	mul32d 8196	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
37	35, 36	eqtr3d 2231	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
38	37	oveq2d 5941	. . . . . 6 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
39		qcn 9725	. . . . . . . 8 |- ( B e. QQ -> B e. CC )
40	7, 39	syl 14	. . . . . . 7 |- ( ph -> B e. CC )
41	14, 34	mulcld 8064	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
42	40, 41, 16	subdird 8458	. . . . . 6 |- ( ph -> ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
43	38, 42	eqtr4d 2232	. . . . 5 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
44	30, 43	eqeq12d 2211	. . . 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 5932	. . . 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 9717	. . . . . . . 8 |- ( C e. ZZ -> C e. QQ )
48	15, 47	syl 14	. . . . . . 7 |- ( ph -> C e. QQ )
49		qmulcl 9728	. . . . . . 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 9471	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( A / D ) ) ) e. ZZ )
52		modqcyc2 10469	. . . . . 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 9728	. . . . . . 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 9471	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( B / D ) ) ) e. ZZ )
57		modqcyc2 10469	. . . . . 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 2211	. . . 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 2167   =/= wne 2367   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   x. cmul 7901   < clt 8078   - cmin 8214   / cdiv 8716   ZZcz 9343   QQcq 9710   |_cfl 10375   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:  modqmul12d  10487  modqnegd  10488  modqmulmod  10498  eulerthlema  12423  fermltl  12427  odzdvds  12439  lgsdir2lem4  15356  lgsdirprm  15359  gausslemma2d  15394