Theorem modsubdir 7521

Description: Distribute the modulo operation over a subtraction.

Assertion

Ref	Expression
modsubdir	|- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((B mod C) <_ (A mod C) <-> ((A - B) mod C) = ((A mod C) - (B mod C))))

Proof of Theorem modsubdir

Step	Hyp	Ref	Expression
1		modcl 7502	. . . 4 |- ((A e. RR /\ C e. RR+) -> (A mod C) e. RR)
2	1	3adant2 895	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (A mod C) e. RR)
3		modcl 7502	. . . 4 |- ((B e. RR /\ C e. RR+) -> (B mod C) e. RR)
4	3	3adant1 894	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (B mod C) e. RR)
5		subge0 6863	. . 3 |- (((A mod C) e. RR /\ (B mod C) e. RR) -> (0 <_ ((A mod C) - (B mod C)) <-> (B mod C) <_ (A mod C)))
6	2, 4, 5	syl11anc 524	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (0 <_ ((A mod C) - (B mod C)) <-> (B mod C) <_ (A mod C)))
7		resubcl 6601	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
8	7	3adant3 896	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (A - B) e. RR)
9		simp3 878	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> C e. RR+)
10		rerpdivcl 7251	. . . . . . . . . 10 |- ((A e. RR /\ C e. RR+) -> (A / C) e. RR)
11		flcl 7465	. . . . . . . . . 10 |- ((A / C) e. RR -> (|_` (A / C)) e. ZZ)
12	10, 11	syl 12	. . . . . . . . 9 |- ((A e. RR /\ C e. RR+) -> (|_` (A / C)) e. ZZ)
13	12	3adant2 895	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (|_` (A / C)) e. ZZ)
14		rerpdivcl 7251	. . . . . . . . . 10 |- ((B e. RR /\ C e. RR+) -> (B / C) e. RR)
15		flcl 7465	. . . . . . . . . 10 |- ((B / C) e. RR -> (|_` (B / C)) e. ZZ)
16	14, 15	syl 12	. . . . . . . . 9 |- ((B e. RR /\ C e. RR+) -> (|_` (B / C)) e. ZZ)
17	16	3adant1 894	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (|_` (B / C)) e. ZZ)
18		zsubcl 7377	. . . . . . . 8 |- (((|_` (A / C)) e. ZZ /\ (|_` (B / C)) e. ZZ) -> ((|_` (A / C)) - (|_` (B / C))) e. ZZ)
19	13, 17, 18	syl11anc 524	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((|_` (A / C)) - (|_` (B / C))) e. ZZ)
20		modcyc2 7517	. . . . . . 7 |- (((A - B) e. RR /\ C e. RR+ /\ ((|_` (A / C)) - (|_` (B / C))) e. ZZ) -> (((A - B) - (C x. ((|_` (A / C)) - (|_` (B / C))))) mod C) = ((A - B) mod C))
21	8, 9, 19, 20	syl111anc 1100	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A - B) - (C x. ((|_` (A / C)) - (|_` (B / C))))) mod C) = ((A - B) mod C))
22		recn 6466	. . . . . . . . . 10 |- (A e. RR -> A e. CC)
23	22	3ad2ant1 897	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> A e. CC)
24		recn 6466	. . . . . . . . . 10 |- (B e. RR -> B e. CC)
25	24	3ad2ant2 898	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> B e. CC)
26		rpre 7236	. . . . . . . . . . . . 13 |- (C e. RR+ -> C e. RR)
27	26	adantl 424	. . . . . . . . . . . 12 |- ((A e. RR /\ C e. RR+) -> C e. RR)
28		reflcl 7466	. . . . . . . . . . . . 13 |- ((A / C) e. RR -> (|_` (A / C)) e. RR)
29	10, 28	syl 12	. . . . . . . . . . . 12 |- ((A e. RR /\ C e. RR+) -> (|_` (A / C)) e. RR)
30		remulcl 6457	. . . . . . . . . . . 12 |- ((C e. RR /\ (|_` (A / C)) e. RR) -> (C x. (|_` (A / C))) e. RR)
31	27, 29, 30	syl11anc 524	. . . . . . . . . . 11 |- ((A e. RR /\ C e. RR+) -> (C x. (|_` (A / C))) e. RR)
32	31	recnd 6468	. . . . . . . . . 10 |- ((A e. RR /\ C e. RR+) -> (C x. (|_` (A / C))) e. CC)
33	32	3adant2 895	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (C x. (|_` (A / C))) e. CC)
34	26	adantl 424	. . . . . . . . . . . 12 |- ((B e. RR /\ C e. RR+) -> C e. RR)
35		reflcl 7466	. . . . . . . . . . . . 13 |- ((B / C) e. RR -> (|_` (B / C)) e. RR)
36	14, 35	syl 12	. . . . . . . . . . . 12 |- ((B e. RR /\ C e. RR+) -> (|_` (B / C)) e. RR)
37		remulcl 6457	. . . . . . . . . . . 12 |- ((C e. RR /\ (|_` (B / C)) e. RR) -> (C x. (|_` (B / C))) e. RR)
38	34, 36, 37	syl11anc 524	. . . . . . . . . . 11 |- ((B e. RR /\ C e. RR+) -> (C x. (|_` (B / C))) e. RR)
39	38	recnd 6468	. . . . . . . . . 10 |- ((B e. RR /\ C e. RR+) -> (C x. (|_` (B / C))) e. CC)
40	39	3adant1 894	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (C x. (|_` (B / C))) e. CC)
41		sub4 6643	. . . . . . . . 9 |- (((A e. CC /\ B e. CC) /\ ((C x. (|_` (A / C))) e. CC /\ (C x. (|_` (B / C))) e. CC)) -> ((A - B) - ((C x. (|_` (A / C))) - (C x. (|_` (B / C))))) = ((A - (C x. (|_` (A / C)))) - (B - (C x. (|_` (B / C))))))
42	23, 25, 33, 40, 41	syl22anc 1101	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A - B) - ((C x. (|_` (A / C))) - (C x. (|_` (B / C))))) = ((A - (C x. (|_` (A / C)))) - (B - (C x. (|_` (B / C))))))
43	26	3ad2ant3 899	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> C e. RR)
44	43	recnd 6468	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> C e. CC)
45	29	recnd 6468	. . . . . . . . . . 11 |- ((A e. RR /\ C e. RR+) -> (|_` (A / C)) e. CC)
46	45	3adant2 895	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (|_` (A / C)) e. CC)
47	36	recnd 6468	. . . . . . . . . . 11 |- ((B e. RR /\ C e. RR+) -> (|_` (B / C)) e. CC)
48	47	3adant1 894	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (|_` (B / C)) e. CC)
49		subdi 6590	. . . . . . . . . 10 |- ((C e. CC /\ (|_` (A / C)) e. CC /\ (|_` (B / C)) e. CC) -> (C x. ((|_` (A / C)) - (|_` (B / C)))) = ((C x. (|_` (A / C))) - (C x. (|_` (B / C)))))
50	44, 46, 48, 49	syl111anc 1100	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (C x. ((|_` (A / C)) - (|_` (B / C)))) = ((C x. (|_` (A / C))) - (C x. (|_` (B / C)))))
51	50	opreq2d 4898	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A - B) - (C x. ((|_` (A / C)) - (|_` (B / C))))) = ((A - B) - ((C x. (|_` (A / C))) - (C x. (|_` (B / C))))))
52		modval 7501	. . . . . . . . . 10 |- ((A e. RR /\ C e. RR+) -> (A mod C) = (A - (C x. (|_` (A / C)))))
53	52	3adant2 895	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (A mod C) = (A - (C x. (|_` (A / C)))))
54		modval 7501	. . . . . . . . . 10 |- ((B e. RR /\ C e. RR+) -> (B mod C) = (B - (C x. (|_` (B / C)))))
55	54	3adant1 894	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (B mod C) = (B - (C x. (|_` (B / C)))))
56	53, 55	opreq12d 4900	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) - (B mod C)) = ((A - (C x. (|_` (A / C)))) - (B - (C x. (|_` (B / C))))))
57	42, 51, 56	3eqtr4d 1937	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A - B) - (C x. ((|_` (A / C)) - (|_` (B / C))))) = ((A mod C) - (B mod C)))
58	57	opreq1d 4897	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A - B) - (C x. ((|_` (A / C)) - (|_` (B / C))))) mod C) = (((A mod C) - (B mod C)) mod C))
59	21, 58	eqtr3d 1927	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A - B) mod C) = (((A mod C) - (B mod C)) mod C))
60	59	adantr 425	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> ((A - B) mod C) = (((A mod C) - (B mod C)) mod C))
61		resubcl 6601	. . . . . . 7 |- (((A mod C) e. RR /\ (B mod C) e. RR) -> ((A mod C) - (B mod C)) e. RR)
62	2, 4, 61	syl11anc 524	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) - (B mod C)) e. RR)
63	62	adantr 425	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> ((A mod C) - (B mod C)) e. RR)
64		simpl3 881	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> C e. RR+)
65		simpr 350	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> 0 <_ ((A mod C) - (B mod C)))
66		modge0 7503	. . . . . . . . 9 |- ((B e. RR /\ C e. RR+) -> 0 <_ (B mod C))
67	66	3adant1 894	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> 0 <_ (B mod C))
68		subge02 6866	. . . . . . . . 9 |- (((A mod C) e. RR /\ (B mod C) e. RR) -> (0 <_ (B mod C) <-> ((A mod C) - (B mod C)) <_ (A mod C)))
69	2, 4, 68	syl11anc 524	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (0 <_ (B mod C) <-> ((A mod C) - (B mod C)) <_ (A mod C)))
70	67, 69	mpbid 212	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) - (B mod C)) <_ (A mod C))
71		modlt 7504	. . . . . . . 8 |- ((A e. RR /\ C e. RR+) -> (A mod C) < C)
72	71	3adant2 895	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (A mod C) < C)
73	62, 2, 43, 70, 72	lelttrd 6697	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) - (B mod C)) < C)
74	73	adantr 425	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> ((A mod C) - (B mod C)) < C)
75		modid 7512	. . . . 5 |- (((((A mod C) - (B mod C)) e. RR /\ C e. RR+) /\ (0 <_ ((A mod C) - (B mod C)) /\ ((A mod C) - (B mod C)) < C)) -> (((A mod C) - (B mod C)) mod C) = ((A mod C) - (B mod C)))
76	63, 64, 65, 74, 75	syl22anc 1101	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> (((A mod C) - (B mod C)) mod C) = ((A mod C) - (B mod C)))
77	60, 76	eqtrd 1925	. . 3 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ 0 <_ ((A mod C) - (B mod C))) -> ((A - B) mod C) = ((A mod C) - (B mod C)))
78		modge0 7503	. . . . . . 7 |- (((A - B) e. RR /\ C e. RR+) -> 0 <_ ((A - B) mod C))
79	78, 7	sylan 497	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ C e. RR+) -> 0 <_ ((A - B) mod C))
80	79	3impa 1062	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> 0 <_ ((A - B) mod C))
81	80	adantr 425	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ ((A - B) mod C) = ((A mod C) - (B mod C))) -> 0 <_ ((A - B) mod C))
82		simpr 350	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ ((A - B) mod C) = ((A mod C) - (B mod C))) -> ((A - B) mod C) = ((A mod C) - (B mod C)))
83	81, 82	breqtrd 3361	. . 3 |- (((A e. RR /\ B e. RR /\ C e. RR+) /\ ((A - B) mod C) = ((A mod C) - (B mod C))) -> 0 <_ ((A mod C) - (B mod C)))
84	77, 83	impbida 577	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (0 <_ ((A mod C) - (B mod C)) <-> ((A - B) mod C) = ((A mod C) - (B mod C))))
85	6, 84	bitr3d 589	1 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((B mod C) <_ (A mod C) <-> ((A - B) mod C) = ((A mod C) - (B mod C))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  ZZcz 6451  RR+crp 6453   < clt 6653  |_cfl 7462   mod cmo 7499

This theorem is referenced by:  digit1 7905

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

Copyright terms: Public domain