Theorem modmul1 7519

Description: Multiplication property of the modulo operation. Note that the multiplier C must be an integer.

Assertion

Ref	Expression
modmul1	|- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+) /\ (A mod D) = (B mod D)) -> ((A x. C) mod D) = ((B x. C) mod D))

Proof of Theorem modmul1

Step	Hyp	Ref	Expression
1		modval 7501	. . . . . . . 8 |- ((A e. RR /\ D e. RR+) -> (A mod D) = (A - (D x. (|_` (A / D)))))
2		modval 7501	. . . . . . . 8 |- ((B e. RR /\ D e. RR+) -> (B mod D) = (B - (D x. (|_` (B / D)))))
3	1, 2	eqeqan12d 1901	. . . . . . 7 |- (((A e. RR /\ D e. RR+) /\ (B e. RR /\ D e. RR+)) -> ((A mod D) = (B mod D) <-> (A - (D x. (|_` (A / D)))) = (B - (D x. (|_` (B / D))))))
4	3	anandirs 571	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ D e. RR+) -> ((A mod D) = (B mod D) <-> (A - (D x. (|_` (A / D)))) = (B - (D x. (|_` (B / D))))))
5	4	adantrl 430	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((A mod D) = (B mod D) <-> (A - (D x. (|_` (A / D)))) = (B - (D x. (|_` (B / D))))))
6		opreq1 4889	. . . . 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))
7	5, 6	syl6bi 231	. . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((A mod D) = (B mod D) -> ((A - (D x. (|_` (A / D)))) x. C) = ((B - (D x. (|_` (B / D)))) x. C)))
8		rpcn 7237	. . . . . . . . . . 11 |- (D e. RR+ -> D e. CC)
9	8	ad2antll 443	. . . . . . . . . 10 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> D e. CC)
10		zcn 7349	. . . . . . . . . . 11 |- (C e. ZZ -> C e. CC)
11	10	ad2antrl 442	. . . . . . . . . 10 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> C e. CC)
12		rerpdivcl 7251	. . . . . . . . . . . . 13 |- ((A e. RR /\ D e. RR+) -> (A / D) e. RR)
13		flcl 7465	. . . . . . . . . . . . 13 |- ((A / D) e. RR -> (|_` (A / D)) e. ZZ)
14	12, 13	syl 12	. . . . . . . . . . . 12 |- ((A e. RR /\ D e. RR+) -> (|_` (A / D)) e. ZZ)
15		zcn 7349	. . . . . . . . . . . 12 |- ((|_` (A / D)) e. ZZ -> (|_` (A / D)) e. CC)
16	14, 15	syl 12	. . . . . . . . . . 11 |- ((A e. RR /\ D e. RR+) -> (|_` (A / D)) e. CC)
17	16	adantrl 430	. . . . . . . . . 10 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (|_` (A / D)) e. CC)
18		mulass 6461	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC /\ (|_` (A / D)) e. CC) -> ((D x. C) x. (|_` (A / D))) = (D x. (C x. (|_` (A / D)))))
19	9, 11, 17, 18	syl111anc 1100	. . . . . . . . 9 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((D x. C) x. (|_` (A / D))) = (D x. (C x. (|_` (A / D)))))
20		mul23 6580	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC /\ (|_` (A / D)) e. CC) -> ((D x. C) x. (|_` (A / D))) = ((D x. (|_` (A / D))) x. C))
21	9, 11, 17, 20	syl111anc 1100	. . . . . . . . 9 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((D x. C) x. (|_` (A / D))) = ((D x. (|_` (A / D))) x. C))
22	19, 21	eqtr3d 1927	. . . . . . . 8 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (D x. (C x. (|_` (A / D)))) = ((D x. (|_` (A / D))) x. C))
23	22	opreq2d 4898	. . . . . . 7 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((A x. C) - (D x. (C x. (|_` (A / D))))) = ((A x. C) - ((D x. (|_` (A / D))) x. C)))
24		recn 6466	. . . . . . . . 9 |- (A e. RR -> A e. CC)
25	24	adantr 425	. . . . . . . 8 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> A e. CC)
26	8	adantl 424	. . . . . . . . . 10 |- ((A e. RR /\ D e. RR+) -> D e. CC)
27		mulcl 6456	. . . . . . . . . 10 |- ((D e. CC /\ (|_` (A / D)) e. CC) -> (D x. (|_` (A / D))) e. CC)
28	26, 16, 27	syl11anc 524	. . . . . . . . 9 |- ((A e. RR /\ D e. RR+) -> (D x. (|_` (A / D))) e. CC)
29	28	adantrl 430	. . . . . . . 8 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (D x. (|_` (A / D))) e. CC)
30		subdir 6591	. . . . . . . 8 |- ((A e. CC /\ (D x. (|_` (A / D))) e. CC /\ C e. CC) -> ((A - (D x. (|_` (A / D)))) x. C) = ((A x. C) - ((D x. (|_` (A / D))) x. C)))
31	25, 29, 11, 30	syl111anc 1100	. . . . . . 7 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((A - (D x. (|_` (A / D)))) x. C) = ((A x. C) - ((D x. (|_` (A / D))) x. C)))
32	23, 31	eqtr4d 1928	. . . . . 6 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((A x. C) - (D x. (C x. (|_` (A / D))))) = ((A - (D x. (|_` (A / D)))) x. C))
33	32	adantlr 429	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((A x. C) - (D x. (C x. (|_` (A / D))))) = ((A - (D x. (|_` (A / D)))) x. C))
34	8	ad2antll 443	. . . . . . . . . 10 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> D e. CC)
35	10	ad2antrl 442	. . . . . . . . . 10 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> C e. CC)
36		rerpdivcl 7251	. . . . . . . . . . . . 13 |- ((B e. RR /\ D e. RR+) -> (B / D) e. RR)
37		flcl 7465	. . . . . . . . . . . . 13 |- ((B / D) e. RR -> (|_` (B / D)) e. ZZ)
38	36, 37	syl 12	. . . . . . . . . . . 12 |- ((B e. RR /\ D e. RR+) -> (|_` (B / D)) e. ZZ)
39		zcn 7349	. . . . . . . . . . . 12 |- ((|_` (B / D)) e. ZZ -> (|_` (B / D)) e. CC)
40	38, 39	syl 12	. . . . . . . . . . 11 |- ((B e. RR /\ D e. RR+) -> (|_` (B / D)) e. CC)
41	40	adantrl 430	. . . . . . . . . 10 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (|_` (B / D)) e. CC)
42		mulass 6461	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC /\ (|_` (B / D)) e. CC) -> ((D x. C) x. (|_` (B / D))) = (D x. (C x. (|_` (B / D)))))
43	34, 35, 41, 42	syl111anc 1100	. . . . . . . . 9 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((D x. C) x. (|_` (B / D))) = (D x. (C x. (|_` (B / D)))))
44		mul23 6580	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC /\ (|_` (B / D)) e. CC) -> ((D x. C) x. (|_` (B / D))) = ((D x. (|_` (B / D))) x. C))
45	34, 35, 41, 44	syl111anc 1100	. . . . . . . . 9 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((D x. C) x. (|_` (B / D))) = ((D x. (|_` (B / D))) x. C))
46	43, 45	eqtr3d 1927	. . . . . . . 8 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (D x. (C x. (|_` (B / D)))) = ((D x. (|_` (B / D))) x. C))
47	46	opreq2d 4898	. . . . . . 7 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((B x. C) - (D x. (C x. (|_` (B / D))))) = ((B x. C) - ((D x. (|_` (B / D))) x. C)))
48		recn 6466	. . . . . . . . 9 |- (B e. RR -> B e. CC)
49	48	adantr 425	. . . . . . . 8 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> B e. CC)
50	8	adantl 424	. . . . . . . . . 10 |- ((B e. RR /\ D e. RR+) -> D e. CC)
51		mulcl 6456	. . . . . . . . . 10 |- ((D e. CC /\ (|_` (B / D)) e. CC) -> (D x. (|_` (B / D))) e. CC)
52	50, 40, 51	syl11anc 524	. . . . . . . . 9 |- ((B e. RR /\ D e. RR+) -> (D x. (|_` (B / D))) e. CC)
53	52	adantrl 430	. . . . . . . 8 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (D x. (|_` (B / D))) e. CC)
54		subdir 6591	. . . . . . . 8 |- ((B e. CC /\ (D x. (|_` (B / D))) e. CC /\ C e. CC) -> ((B - (D x. (|_` (B / D)))) x. C) = ((B x. C) - ((D x. (|_` (B / D))) x. C)))
55	49, 53, 35, 54	syl111anc 1100	. . . . . . 7 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((B - (D x. (|_` (B / D)))) x. C) = ((B x. C) - ((D x. (|_` (B / D))) x. C)))
56	47, 55	eqtr4d 1928	. . . . . 6 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> ((B x. C) - (D x. (C x. (|_` (B / D))))) = ((B - (D x. (|_` (B / D)))) x. C))
57	56	adantll 428	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((B x. C) - (D x. (C x. (|_` (B / D))))) = ((B - (D x. (|_` (B / D)))) x. C))
58	33, 57	eqeq12d 1899	. . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> (((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)))
59	7, 58	sylibrd 221	. . 3 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((A mod D) = (B mod D) -> ((A x. C) - (D x. (C x. (|_` (A / D))))) = ((B x. C) - (D x. (C x. (|_` (B / D)))))))
60		remulcl 6457	. . . . . . . . 9 |- ((A e. RR /\ C e. RR) -> (A x. C) e. RR)
61		zre 7348	. . . . . . . . 9 |- (C e. ZZ -> C e. RR)
62	60, 61	sylan2 500	. . . . . . . 8 |- ((A e. RR /\ C e. ZZ) -> (A x. C) e. RR)
63	62	adantrr 431	. . . . . . 7 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (A x. C) e. RR)
64		simprr 451	. . . . . . 7 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> D e. RR+)
65		simprl 450	. . . . . . . 8 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> C e. ZZ)
66	14	adantrl 430	. . . . . . . 8 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (|_` (A / D)) e. ZZ)
67		zmulcl 7389	. . . . . . . 8 |- ((C e. ZZ /\ (|_` (A / D)) e. ZZ) -> (C x. (|_` (A / D))) e. ZZ)
68	65, 66, 67	syl11anc 524	. . . . . . 7 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (C x. (|_` (A / D))) e. ZZ)
69		modcyc2 7517	. . . . . . 7 |- (((A x. C) e. RR /\ D e. RR+ /\ (C x. (|_` (A / D))) e. ZZ) -> (((A x. C) - (D x. (C x. (|_` (A / D))))) mod D) = ((A x. C) mod D))
70	63, 64, 68, 69	syl111anc 1100	. . . . . 6 |- ((A e. RR /\ (C e. ZZ /\ D e. RR+)) -> (((A x. C) - (D x. (C x. (|_` (A / D))))) mod D) = ((A x. C) mod D))
71	70	adantlr 429	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> (((A x. C) - (D x. (C x. (|_` (A / D))))) mod D) = ((A x. C) mod D))
72		remulcl 6457	. . . . . . . . 9 |- ((B e. RR /\ C e. RR) -> (B x. C) e. RR)
73	72, 61	sylan2 500	. . . . . . . 8 |- ((B e. RR /\ C e. ZZ) -> (B x. C) e. RR)
74	73	adantrr 431	. . . . . . 7 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (B x. C) e. RR)
75		simprr 451	. . . . . . 7 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> D e. RR+)
76		simprl 450	. . . . . . . 8 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> C e. ZZ)
77	38	adantrl 430	. . . . . . . 8 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (|_` (B / D)) e. ZZ)
78		zmulcl 7389	. . . . . . . 8 |- ((C e. ZZ /\ (|_` (B / D)) e. ZZ) -> (C x. (|_` (B / D))) e. ZZ)
79	76, 77, 78	syl11anc 524	. . . . . . 7 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (C x. (|_` (B / D))) e. ZZ)
80		modcyc2 7517	. . . . . . 7 |- (((B x. C) e. RR /\ D e. RR+ /\ (C x. (|_` (B / D))) e. ZZ) -> (((B x. C) - (D x. (C x. (|_` (B / D))))) mod D) = ((B x. C) mod D))
81	74, 75, 79, 80	syl111anc 1100	. . . . . 6 |- ((B e. RR /\ (C e. ZZ /\ D e. RR+)) -> (((B x. C) - (D x. (C x. (|_` (B / D))))) mod D) = ((B x. C) mod D))
82	81	adantll 428	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> (((B x. C) - (D x. (C x. (|_` (B / D))))) mod D) = ((B x. C) mod D))
83	71, 82	eqeq12d 1899	. . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((((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)))
84		opreq1 4889	. . . 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))
85	83, 84	syl5bi 225	. . 3 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> (((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)))
86	59, 85	syld 30	. 2 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+)) -> ((A mod D) = (B mod D) -> ((A x. C) mod D) = ((B x. C) mod D)))
87	86	3impia 1064	1 |- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+) /\ (A mod D) = (B mod D)) -> ((A x. C) mod D) = ((B x. C) mod D))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385   x. cmul 6391   - cmin 6445   / cdiv 6447  ZZcz 6451  RR+crp 6453  |_cfl 7462   mod cmo 7499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500

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