Theorem modqmul1 10594

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 10541	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1271	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqmul1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10541	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1271	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2244	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 6007	. . . . 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 9825	. . . . . . . . . 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 9566	. . . . . . . . 9 |- ( ph -> C e. CC )
17	4	gt0ne0d 8655	. . . . . . . . . . . 12 |- ( ph -> D =/= 0 )
18		qdivcl 9834	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
19	2, 3, 17, 18	syl3anc 1271	. . . . . . . . . . 11 |- ( ph -> ( A / D ) e. QQ )
20	19	flqcld 10492	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
21	20	zcnd 9566	. . . . . . . . 9 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
22	14, 16, 21	mulassd 8166	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) )
23	14, 16, 21	mul32d 8295	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( A / D ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
24	22, 23	eqtr3d 2264	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) = ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) )
25	24	oveq2d 6016	. . . . . 6 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
26		qcn 9825	. . . . . . . 8 |- ( A e. QQ -> A e. CC )
27	2, 26	syl 14	. . . . . . 7 |- ( ph -> A e. CC )
28	14, 21	mulcld 8163	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
29	27, 28, 16	subdird 8557	. . . . . 6 |- ( ph -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) = ( ( A x. C ) - ( ( D x. ( |_ ` ( A / D ) ) ) x. C ) ) )
30	25, 29	eqtr4d 2265	. . . . 5 |- ( ph -> ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) x. C ) )
31		qdivcl 9834	. . . . . . . . . . . 12 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
32	7, 3, 17, 31	syl3anc 1271	. . . . . . . . . . 11 |- ( ph -> ( B / D ) e. QQ )
33	32	flqcld 10492	. . . . . . . . . 10 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
34	33	zcnd 9566	. . . . . . . . 9 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
35	14, 16, 34	mulassd 8166	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) )
36	14, 16, 34	mul32d 8295	. . . . . . . 8 |- ( ph -> ( ( D x. C ) x. ( |_ ` ( B / D ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
37	35, 36	eqtr3d 2264	. . . . . . 7 |- ( ph -> ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) = ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) )
38	37	oveq2d 6016	. . . . . 6 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
39		qcn 9825	. . . . . . . 8 |- ( B e. QQ -> B e. CC )
40	7, 39	syl 14	. . . . . . 7 |- ( ph -> B e. CC )
41	14, 34	mulcld 8163	. . . . . . 7 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
42	40, 41, 16	subdird 8557	. . . . . 6 |- ( ph -> ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) = ( ( B x. C ) - ( ( D x. ( |_ ` ( B / D ) ) ) x. C ) ) )
43	38, 42	eqtr4d 2265	. . . . 5 |- ( ph -> ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) x. C ) )
44	30, 43	eqeq12d 2244	. . . 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 6007	. . . 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 9817	. . . . . . . 8 |- ( C e. ZZ -> C e. QQ )
48	15, 47	syl 14	. . . . . . 7 |- ( ph -> C e. QQ )
49		qmulcl 9828	. . . . . . 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 9571	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( A / D ) ) ) e. ZZ )
52		modqcyc2 10577	. . . . . 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 1272	. . . . 5 |- ( ph -> ( ( ( A x. C ) - ( D x. ( C x. ( |_ ` ( A / D ) ) ) ) ) mod D ) = ( ( A x. C ) mod D ) )
54		qmulcl 9828	. . . . . . 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 9571	. . . . . 6 |- ( ph -> ( C x. ( |_ ` ( B / D ) ) ) e. ZZ )
57		modqcyc2 10577	. . . . . 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 1272	. . . . 5 |- ( ph -> ( ( ( B x. C ) - ( D x. ( C x. ( |_ ` ( B / D ) ) ) ) ) mod D ) = ( ( B x. C ) mod D ) )
59	53, 58	eqeq12d 2244	. . . 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 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   0cc0 7995   x. cmul 8000   < clt 8177   - cmin 8313   / cdiv 8815   ZZcz 9442   QQcq 9810   |_cfl 10483   mod cmo 10539

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-fl 10485  df-mod 10540

This theorem is referenced by:  modqmul12d  10595  modqnegd  10596  modqmulmod  10606  eulerthlema  12747  fermltl  12751  odzdvds  12763  lgsdir2lem4  15704  lgsdirprm  15707  gausslemma2d  15742