Theorem modqadd1 10798

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 10761	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1278	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqadd1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10761	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1278	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2253	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 6092	. . . . 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	biimtrdi 163	. . . 4 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) )
13		qcn 10034	. . . . . . 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 10034	. . . . . . 7 |- ( C e. QQ -> C e. CC )
17	15, 16	syl 14	. . . . . 6 |- ( ph -> C e. CC )
18		qcn 10034	. . . . . . . 8 |- ( D e. QQ -> D e. CC )
19	3, 18	syl 14	. . . . . . 7 |- ( ph -> D e. CC )
20	4	gt0ne0d 8840	. . . . . . . . . 10 |- ( ph -> D =/= 0 )
21		qdivcl 10043	. . . . . . . . . 10 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
22	2, 3, 20, 21	syl3anc 1278	. . . . . . . . 9 |- ( ph -> ( A / D ) e. QQ )
23	22	flqcld 10712	. . . . . . . 8 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
24	23	zcnd 9769	. . . . . . 7 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
25	19, 24	mulcld 8346	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
26	14, 17, 25	addsubd 8658	. . . . 5 |- ( ph -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) )
27		qcn 10034	. . . . . . 7 |- ( B e. QQ -> B e. CC )
28	7, 27	syl 14	. . . . . 6 |- ( ph -> B e. CC )
29		qdivcl 10043	. . . . . . . . . 10 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
30	7, 3, 20, 29	syl3anc 1278	. . . . . . . . 9 |- ( ph -> ( B / D ) e. QQ )
31	30	flqcld 10712	. . . . . . . 8 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
32	31	zcnd 9769	. . . . . . 7 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
33	19, 32	mulcld 8346	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
34	28, 17, 33	addsubd 8658	. . . . 5 |- ( ph -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
35	26, 34	eqeq12d 2253	. . . 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 169	. . 3 |- ( ph -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
37		oveq1 6092	. . . 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 10035	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A + C ) e. QQ )
39	2, 15, 38	syl2anc 415	. . . . . 6 |- ( ph -> ( A + C ) e. QQ )
40		modqcyc2 10797	. . . . . 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 1279	. . . . 5 |- ( ph -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
42		qaddcl 10035	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B + C ) e. QQ )
43	7, 15, 42	syl2anc 415	. . . . . 6 |- ( ph -> ( B + C ) e. QQ )
44		modqcyc2 10797	. . . . . 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 1279	. . . . 5 |- ( ph -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
46	41, 45	eqeq12d 2253	. . . 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	imbitrid 154	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   ZZcz 9644   QQcq 10019   |_cfl 10703   mod cmo 10759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760

This theorem is used by:  modqaddabs  10799  modqaddmod  10800  modqadd12d  10817  modqaddmulmod  10828  moddvds  12566  modsubi  13198  lgsvalmod  16138  lgsmod  16145  lgsne0  16157  lgseisen  16193