Theorem modqmul1 10333

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 10280	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1233	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqmul1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10280	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1233	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2185	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 5860	. . . . 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 162	. . . 4 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) ) )
13		qcn 9593	. . . . . . . . . 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 9335	. . . . . . . . 9 |- ( ph -> C e. CC )
17	4	gt0ne0d 8431	. . . . . . . . . . . 12 |- ( ph -> D =/= 0 )
18		qdivcl 9602	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
19	2, 3, 17, 18	syl3anc 1233	. . . . . . . . . . 11 |- ( ph -> ( A / D ) e. QQ )
20	19	flqcld 10233	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
21	20	zcnd 9335	. . . . . . . . 9 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
22	14, 16, 21	mulassd 7943	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) )
23	14, 16, 21	mul32d 8072	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
24	22, 23	eqtr3d 2205	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
25	24	oveq2d 5869	. . . . . 6 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
26		qcn 9593	. . . . . . . 8 |- ( A e. QQ -> A e. CC )
27	2, 26	syl 14	. . . . . . 7 |- ( ph -> A e. CC )
28	14, 21	mulcld 7940	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
29	27, 28, 16	subdird 8334	. . . . . 6 |- ( ph -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
30	25, 29	eqtr4d 2206	. . . . 5 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) )
31		qdivcl 9602	. . . . . . . . . . . 12 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
32	7, 3, 17, 31	syl3anc 1233	. . . . . . . . . . 11 |- ( ph -> ( B / D ) e. QQ )
33	32	flqcld 10233	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
34	33	zcnd 9335	. . . . . . . . 9 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
35	14, 16, 34	mulassd 7943	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) )
36	14, 16, 34	mul32d 8072	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
37	35, 36	eqtr3d 2205	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
38	37	oveq2d 5869	. . . . . 6 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
39		qcn 9593	. . . . . . . 8 |- ( B e. QQ -> B e. CC )
40	7, 39	syl 14	. . . . . . 7 |- ( ph -> B e. CC )
41	14, 34	mulcld 7940	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
42	40, 41, 16	subdird 8334	. . . . . 6 |- ( ph -> ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
43	38, 42	eqtr4d 2206	. . . . 5 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
44	30, 43	eqeq12d 2185	. . . 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 168	. . 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 5860	. . . 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 9585	. . . . . . . 8 |- ( C e. ZZ -> C e. QQ )
48	15, 47	syl 14	. . . . . . 7 |- ( ph -> C e. QQ )
49		qmulcl 9596	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A x. C ) e. QQ )
50	2, 48, 49	syl2anc 409	. . . . . 6 |- ( ph -> ( A x. C ) e. QQ )
51	15, 20	zmulcld 9340	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( A / D ) ) ) e. ZZ )
52		modqcyc2 10316	. . . . . 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 1234	. . . . 5 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( A x. C ) mod D ) )
54		qmulcl 9596	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
55	7, 48, 54	syl2anc 409	. . . . . 6 |- ( ph -> ( B x. C ) e. QQ )
56	15, 33	zmulcld 9340	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( B / D ) ) ) e. ZZ )
57		modqcyc2 10316	. . . . . 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 1234	. . . . 5 |- ( ph -> ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) = ( ( B x. C ) mod D ) )
59	53, 58	eqeq12d 2185	. . . 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 153	. . 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 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   x. cmul 7779   < clt 7954   - cmin 8090   / cdiv 8589   ZZcz 9212   QQcq 9578   |_cfl 10224   mod cmo 10278

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226  df-mod 10279

This theorem is referenced by:  modqmul12d  10334  modqnegd  10335  modqmulmod  10345  eulerthlema  12184  fermltl  12188  odzdvds  12199  lgsdir2lem4  13726  lgsdirprm  13729