Theorem modadd1 12707

Description: Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)

Assertion

Ref	Expression
modadd1	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )

Proof of Theorem modadd1

Step	Hyp	Ref	Expression
1		modval 12670	. . . . . . . 8 |- ( ( A e. RR /\ D e. RR+ ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
2		modval 12670	. . . . . . . 8 |- ( ( B e. RR /\ D e. RR+ ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
3	1, 2	eqeqan12d 2638	. . . . . . 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 874	. . . . . 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 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
6		oveq1 6657	. . . . 5 |- ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
7	5, 6	syl6bi 243	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) )
8		recn 10026	. . . . . . . 8 |- ( A e. RR -> A e. CC )
9	8	adantr 481	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> A e. CC )
10		recn 10026	. . . . . . . 8 |- ( C e. RR -> C e. CC )
11	10	ad2antrl 764	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> C e. CC )
12		rpcn 11841	. . . . . . . . . 10 |- ( D e. RR+ -> D e. CC )
13	12	adantl 482	. . . . . . . . 9 |- ( ( A e. RR /\ D e. RR+ ) -> D e. CC )
14		rerpdivcl 11861	. . . . . . . . . 10 |- ( ( A e. RR /\ D e. RR+ ) -> ( A / D ) e. RR )
15		reflcl 12597	. . . . . . . . . . 11 |- ( ( A / D ) e. RR -> ( |_ ` ( A / D ) ) e. RR )
16	15	recnd 10068	. . . . . . . . . 10 |- ( ( A / D ) e. RR -> ( |_ ` ( A / D ) ) e. CC )
17	14, 16	syl 17	. . . . . . . . 9 |- ( ( A e. RR /\ D e. RR+ ) -> ( |_ ` ( A / D ) ) e. CC )
18	13, 17	mulcld 10060	. . . . . . . 8 |- ( ( A e. RR /\ D e. RR+ ) -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
19	18	adantrl 752	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
20	9, 11, 19	addsubd 10413	. . . . . 6 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) )
21	20	adantlr 751	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) )
22		recn 10026	. . . . . . . 8 |- ( B e. RR -> B e. CC )
23	22	adantr 481	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> B e. CC )
24	10	ad2antrl 764	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> C e. CC )
25	12	adantl 482	. . . . . . . . 9 |- ( ( B e. RR /\ D e. RR+ ) -> D e. CC )
26		rerpdivcl 11861	. . . . . . . . . 10 |- ( ( B e. RR /\ D e. RR+ ) -> ( B / D ) e. RR )
27		reflcl 12597	. . . . . . . . . . 11 |- ( ( B / D ) e. RR -> ( |_ ` ( B / D ) ) e. RR )
28	27	recnd 10068	. . . . . . . . . 10 |- ( ( B / D ) e. RR -> ( |_ ` ( B / D ) ) e. CC )
29	26, 28	syl 17	. . . . . . . . 9 |- ( ( B e. RR /\ D e. RR+ ) -> ( |_ ` ( B / D ) ) e. CC )
30	25, 29	mulcld 10060	. . . . . . . 8 |- ( ( B e. RR /\ D e. RR+ ) -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
31	30	adantrl 752	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
32	23, 24, 31	addsubd 10413	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
33	32	adantll 750	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
34	21, 33	eqeq12d 2637	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) <-> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) )
35	7, 34	sylibrd 249	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
36		oveq1 6657	. . . 4 |- ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) )
37		readdcl 10019	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
38	37	adantrr 753	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( A + C ) e. RR )
39		simprr 796	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> D e. RR+ )
40	14	flcld 12599	. . . . . . . 8 |- ( ( A e. RR /\ D e. RR+ ) -> ( |_ ` ( A / D ) ) e. ZZ )
41	40	adantrl 752	. . . . . . 7 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( |_ ` ( A / D ) ) e. ZZ )
42		modcyc2 12706	. . . . . . 7 |- ( ( ( A + C ) e. RR /\ D e. RR+ /\ ( |_ ` ( A / D ) ) e. ZZ ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
43	38, 39, 41, 42	syl3anc 1326	. . . . . 6 |- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
44	43	adantlr 751	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
45		readdcl 10019	. . . . . . . 8 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
46	45	adantrr 753	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( B + C ) e. RR )
47		simprr 796	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> D e. RR+ )
48	26	flcld 12599	. . . . . . . 8 |- ( ( B e. RR /\ D e. RR+ ) -> ( |_ ` ( B / D ) ) e. ZZ )
49	48	adantrl 752	. . . . . . 7 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( |_ ` ( B / D ) ) e. ZZ )
50		modcyc2 12706	. . . . . . 7 |- ( ( ( B + C ) e. RR /\ D e. RR+ /\ ( |_ ` ( B / D ) ) e. ZZ ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
51	46, 47, 49, 50	syl3anc 1326	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
52	51	adantll 750	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
53	44, 52	eqeq12d 2637	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) <-> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
54	36, 53	syl5ib 234	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
55	35, 54	syld 47	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
56	55	3impia 1261	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   x. cmul 9941   - 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:  modaddabs  12708  modaddmod  12709  modadd12d  12726  modaddmulmod  12737  moddvds  14991  modsubi  15776  lgslem4  25025  lgsvalmod  25041  lgsmod  25048  lgsne0  25060  lgseisen  25104  pellexlem6  37398