Theorem modsubdir 12739

Description: Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)

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 12672	. . . 4 |- ( ( A e. RR /\ C e. RR+ ) -> ( A mod C ) e. RR )
2	1	3adant2 1080	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( A mod C ) e. RR )
3		modcl 12672	. . . 4 |- ( ( B e. RR /\ C e. RR+ ) -> ( B mod C ) e. RR )
4	3	3adant1 1079	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( B mod C ) e. RR )
5	2, 4	subge0d 10617	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( 0 <_ ( ( A mod C ) - ( B mod C ) ) <-> ( B mod C ) <_ ( A mod C ) ) )
6		resubcl 10345	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
7	6	3adant3 1081	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( A - B ) e. RR )
8		simp3 1063	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> C e. RR+ )
9		rerpdivcl 11861	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR+ ) -> ( A / C ) e. RR )
10	9	flcld 12599	. . . . . . . . 9 |- ( ( A e. RR /\ C e. RR+ ) -> ( |_ ` ( A / C ) ) e. ZZ )
11	10	3adant2 1080	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( |_ ` ( A / C ) ) e. ZZ )
12		rerpdivcl 11861	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR+ ) -> ( B / C ) e. RR )
13	12	flcld 12599	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR+ ) -> ( |_ ` ( B / C ) ) e. ZZ )
14	13	3adant1 1079	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( |_ ` ( B / C ) ) e. ZZ )
15	11, 14	zsubcld 11487	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( |_ ` ( A / C ) ) - ( |_ ` ( B / C ) ) ) e. ZZ )
16		modcyc2 12706	. . . . . . 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 ) )
17	7, 8, 15, 16	syl3anc 1326	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( ( A - B ) - ( C x. ( ( |_ ` ( A / C ) ) - ( |_ ` ( B / C ) ) ) ) ) mod C ) = ( ( A - B ) mod C ) )
18		recn 10026	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
19	18	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> A e. CC )
20		recn 10026	. . . . . . . . . 10 |- ( B e. RR -> B e. CC )
21	20	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> B e. CC )
22		rpre 11839	. . . . . . . . . . . . 13 |- ( C e. RR+ -> C e. RR )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. RR /\ C e. RR+ ) -> C e. RR )
24		refldivcl 12624	. . . . . . . . . . . 12 |- ( ( A e. RR /\ C e. RR+ ) -> ( |_ ` ( A / C ) ) e. RR )
25	23, 24	remulcld 10070	. . . . . . . . . . 11 |- ( ( A e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( A / C ) ) ) e. RR )
26	25	recnd 10068	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( A / C ) ) ) e. CC )
27	26	3adant2 1080	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( A / C ) ) ) e. CC )
28	22	adantl 482	. . . . . . . . . . . 12 |- ( ( B e. RR /\ C e. RR+ ) -> C e. RR )
29		refldivcl 12624	. . . . . . . . . . . 12 |- ( ( B e. RR /\ C e. RR+ ) -> ( |_ ` ( B / C ) ) e. RR )
30	28, 29	remulcld 10070	. . . . . . . . . . 11 |- ( ( B e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. RR )
31	30	recnd 10068	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. CC )
32	31	3adant1 1079	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( C x. ( |_ ` ( B / C ) ) ) e. CC )
33	19, 21, 27, 32	sub4d 10441	. . . . . . . 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 ) ) ) ) ) )
34	22	3ad2ant3 1084	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> C e. RR )
35	34	recnd 10068	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> C e. CC )
36	24	recnd 10068	. . . . . . . . . . 11 |- ( ( A e. RR /\ C e. RR+ ) -> ( |_ ` ( A / C ) ) e. CC )
37	36	3adant2 1080	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( |_ ` ( A / C ) ) e. CC )
38	29	recnd 10068	. . . . . . . . . . 11 |- ( ( B e. RR /\ C e. RR+ ) -> ( |_ ` ( B / C ) ) e. CC )
39	38	3adant1 1079	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( |_ ` ( B / C ) ) e. CC )
40	35, 37, 39	subdid 10486	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( C x. ( ( |_ ` ( A / C ) ) - ( |_ ` ( B / C ) ) ) ) = ( ( C x. ( |_ ` ( A / C ) ) ) - ( C x. ( |_ ` ( B / C ) ) ) ) )
41	40	oveq2d 6666	. . . . . . . 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 ) ) ) ) ) )
42		modval 12670	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR+ ) -> ( A mod C ) = ( A - ( C x. ( |_ ` ( A / C ) ) ) ) )
43	42	3adant2 1080	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( A mod C ) = ( A - ( C x. ( |_ ` ( A / C ) ) ) ) )
44		modval 12670	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR+ ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
45	44	3adant1 1079	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( B mod C ) = ( B - ( C x. ( |_ ` ( B / C ) ) ) ) )
46	43, 45	oveq12d 6668	. . . . . . . 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 ) ) ) ) ) )
47	33, 41, 46	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( A - B ) - ( C x. ( ( |_ ` ( A / C ) ) - ( |_ ` ( B / C ) ) ) ) ) = ( ( A mod C ) - ( B mod C ) ) )
48	47	oveq1d 6665	. . . . . 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 ) )
49	17, 48	eqtr3d 2658	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( A - B ) mod C ) = ( ( ( A mod C ) - ( B mod C ) ) mod C ) )
50	49	adantr 481	. . . 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 ) )
51	2, 4	resubcld 10458	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( A mod C ) - ( B mod C ) ) e. RR )
52	51	adantr 481	. . . . 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 )
53		simpl3 1066	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR+ ) /\ 0 <_ ( ( A mod C ) - ( B mod C ) ) ) -> C e. RR+ )
54		simpr 477	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR+ ) /\ 0 <_ ( ( A mod C ) - ( B mod C ) ) ) -> 0 <_ ( ( A mod C ) - ( B mod C ) ) )
55		modge0 12678	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR+ ) -> 0 <_ ( B mod C ) )
56	55	3adant1 1079	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> 0 <_ ( B mod C ) )
57	2, 4	subge02d 10619	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( 0 <_ ( B mod C ) <-> ( ( A mod C ) - ( B mod C ) ) <_ ( A mod C ) ) )
58	56, 57	mpbid 222	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( A mod C ) - ( B mod C ) ) <_ ( A mod C ) )
59		modlt 12679	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR+ ) -> ( A mod C ) < C )
60	59	3adant2 1080	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( A mod C ) < C )
61	51, 2, 34, 58, 60	lelttrd 10195	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( A mod C ) - ( B mod C ) ) < C )
62	61	adantr 481	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR+ ) /\ 0 <_ ( ( A mod C ) - ( B mod C ) ) ) -> ( ( A mod C ) - ( B mod C ) ) < C )
63		modid 12695	. . . . 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 ) ) )
64	52, 53, 54, 62, 63	syl22anc 1327	. . . 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 ) ) )
65	50, 64	eqtrd 2656	. . 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 ) ) )
66		modge0 12678	. . . . . 6 |- ( ( ( A - B ) e. RR /\ C e. RR+ ) -> 0 <_ ( ( A - B ) mod C ) )
67	6, 66	stoic3 1701	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> 0 <_ ( ( A - B ) mod C ) )
68	67	adantr 481	. . . 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 ) )
69		simpr 477	. . . 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 ) ) )
70	68, 69	breqtrd 4679	. . 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 ) ) )
71	65, 70	impbida 877	. 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 ) ) ) )
72	5, 71	bitr3d 270	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 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   ZZcz 11377   RR+crp 11832   |_cfl 12591   mod cmo 12668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669

This theorem is referenced by:  modeqmodmin  12740  digit1  12998  4sqlem12  15660