Theorem modqadd1 10260

Description: Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)

Hypotheses

Ref	Expression
modqadd1.a	|- ( ph -> A e. QQ )
modqadd1.b	|- ( ph -> B e. QQ )
modqadd1.c	|- ( ph -> C e. QQ )
modqadd1.dq	|- ( ph -> D e. QQ )
modqadd1.dgt0	|- ( ph -> 0 < D )
modqadd1.ab	|- ( ph -> ( A mod D ) = ( B mod D ) )

Assertion

Ref	Expression
modqadd1	|- ( ph -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )

Proof of Theorem modqadd1

Step	Hyp	Ref	Expression
1		modqadd1.ab	. 2 |- ( ph -> ( A mod D ) = ( B mod D ) )
2		modqadd1.a	. . . . . . 7 |- ( ph -> A e. QQ )
3		modqadd1.dq	. . . . . . 7 |- ( ph -> D e. QQ )
4		modqadd1.dgt0	. . . . . . 7 |- ( ph -> 0 < D )
5		modqval 10223	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1220	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqadd1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10223	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1220	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2172	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 5831	. . . . 5 |- ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
12	10, 11	syl6bi 162	. . . 4 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) )
13		qcn 9543	. . . . . . 7 |- ( A e. QQ -> A e. CC )
14	2, 13	syl 14	. . . . . 6 |- ( ph -> A e. CC )
15		modqadd1.c	. . . . . . 7 |- ( ph -> C e. QQ )
16		qcn 9543	. . . . . . 7 |- ( C e. QQ -> C e. CC )
17	15, 16	syl 14	. . . . . 6 |- ( ph -> C e. CC )
18		qcn 9543	. . . . . . . 8 |- ( D e. QQ -> D e. CC )
19	3, 18	syl 14	. . . . . . 7 |- ( ph -> D e. CC )
20	4	gt0ne0d 8387	. . . . . . . . . 10 |- ( ph -> D =/= 0 )
21		qdivcl 9552	. . . . . . . . . 10 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
22	2, 3, 20, 21	syl3anc 1220	. . . . . . . . 9 |- ( ph -> ( A / D ) e. QQ )
23	22	flqcld 10176	. . . . . . . 8 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
24	23	zcnd 9287	. . . . . . 7 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
25	19, 24	mulcld 7898	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
26	14, 17, 25	addsubd 8207	. . . . 5 |- ( ph -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) )
27		qcn 9543	. . . . . . 7 |- ( B e. QQ -> B e. CC )
28	7, 27	syl 14	. . . . . 6 |- ( ph -> B e. CC )
29		qdivcl 9552	. . . . . . . . . 10 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
30	7, 3, 20, 29	syl3anc 1220	. . . . . . . . 9 |- ( ph -> ( B / D ) e. QQ )
31	30	flqcld 10176	. . . . . . . 8 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
32	31	zcnd 9287	. . . . . . 7 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
33	19, 32	mulcld 7898	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
34	28, 17, 33	addsubd 8207	. . . . 5 |- ( ph -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
35	26, 34	eqeq12d 2172	. . . 4 |- ( ph -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) <-> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) )
36	12, 35	sylibrd 168	. . 3 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
37		oveq1 5831	. . . 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 ) )
38		qaddcl 9544	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A + C ) e. QQ )
39	2, 15, 38	syl2anc 409	. . . . . 6 |- ( ph -> ( A + C ) e. QQ )
40		modqcyc2 10259	. . . . . 6 |- ( ( ( ( A + C ) e. QQ /\ ( |_ ` ( A / D ) ) e. ZZ ) /\ ( D e. QQ /\ 0 < D ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
41	39, 23, 3, 4, 40	syl22anc 1221	. . . . 5 |- ( ph -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
42		qaddcl 9544	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B + C ) e. QQ )
43	7, 15, 42	syl2anc 409	. . . . . 6 |- ( ph -> ( B + C ) e. QQ )
44		modqcyc2 10259	. . . . . 6 |- ( ( ( ( B + C ) e. QQ /\ ( |_ ` ( B / D ) ) e. ZZ ) /\ ( D e. QQ /\ 0 < D ) ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
45	43, 31, 3, 4, 44	syl22anc 1221	. . . . 5 |- ( ph -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
46	41, 45	eqeq12d 2172	. . . 4 |- ( ph -> ( ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) <-> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
47	37, 46	syl5ib 153	. . 3 |- ( ph -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
48	36, 47	syld 45	. 2 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) )
49	1, 48	mpd 13	1 |- ( ph -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   =/= wne 2327   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732   + caddc 7735   x. cmul 7737   < clt 7912   - cmin 8046   / cdiv 8545   ZZcz 9167   QQcq 9528   |_cfl 10167   mod cmo 10221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-q 9529  df-rp 9561  df-fl 10169  df-mod 10222

This theorem is referenced by:  modqaddabs  10261  modqaddmod  10262  modqadd12d  10279  modqaddmulmod  10290  moddvds  11695