Theorem modqadd1 10686

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 10649	. . . . . . 7 |- ( ( A e. QQ /\ D e. QQ /\ 0 < D ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
6	2, 3, 4, 5	syl3anc 1274	. . . . . 6 |- ( ph -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) )
7		modqadd1.b	. . . . . . 7 |- ( ph -> B e. QQ )
8		modqval 10649	. . . . . . 7 |- ( ( B e. QQ /\ D e. QQ /\ 0 < D ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
9	7, 3, 4, 8	syl3anc 1274	. . . . . 6 |- ( ph -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) )
10	6, 9	eqeq12d 2246	. . . . 5 |- ( ph -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) )
11		oveq1 6035	. . . . 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 9929	. . . . . . 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 9929	. . . . . . 7 |- ( C e. QQ -> C e. CC )
17	15, 16	syl 14	. . . . . 6 |- ( ph -> C e. CC )
18		qcn 9929	. . . . . . . 8 |- ( D e. QQ -> D e. CC )
19	3, 18	syl 14	. . . . . . 7 |- ( ph -> D e. CC )
20	4	gt0ne0d 8751	. . . . . . . . . 10 |- ( ph -> D =/= 0 )
21		qdivcl 9938	. . . . . . . . . 10 |- ( ( A e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( A / D ) e. QQ )
22	2, 3, 20, 21	syl3anc 1274	. . . . . . . . 9 |- ( ph -> ( A / D ) e. QQ )
23	22	flqcld 10600	. . . . . . . 8 |- ( ph -> ( |_ ` ( A / D ) ) e. ZZ )
24	23	zcnd 9664	. . . . . . 7 |- ( ph -> ( |_ ` ( A / D ) ) e. CC )
25	19, 24	mulcld 8259	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( A / D ) ) ) e. CC )
26	14, 17, 25	addsubd 8570	. . . . 5 |- ( ph -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) )
27		qcn 9929	. . . . . . 7 |- ( B e. QQ -> B e. CC )
28	7, 27	syl 14	. . . . . 6 |- ( ph -> B e. CC )
29		qdivcl 9938	. . . . . . . . . 10 |- ( ( B e. QQ /\ D e. QQ /\ D =/= 0 ) -> ( B / D ) e. QQ )
30	7, 3, 20, 29	syl3anc 1274	. . . . . . . . 9 |- ( ph -> ( B / D ) e. QQ )
31	30	flqcld 10600	. . . . . . . 8 |- ( ph -> ( |_ ` ( B / D ) ) e. ZZ )
32	31	zcnd 9664	. . . . . . 7 |- ( ph -> ( |_ ` ( B / D ) ) e. CC )
33	19, 32	mulcld 8259	. . . . . 6 |- ( ph -> ( D x. ( |_ ` ( B / D ) ) ) e. CC )
34	28, 17, 33	addsubd 8570	. . . . 5 |- ( ph -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) )
35	26, 34	eqeq12d 2246	. . . 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 6035	. . . 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 9930	. . . . . . 7 |- ( ( A e. QQ /\ C e. QQ ) -> ( A + C ) e. QQ )
39	2, 15, 38	syl2anc 411	. . . . . 6 |- ( ph -> ( A + C ) e. QQ )
40		modqcyc2 10685	. . . . . 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 1275	. . . . 5 |- ( ph -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) )
42		qaddcl 9930	. . . . . . 7 |- ( ( B e. QQ /\ C e. QQ ) -> ( B + C ) e. QQ )
43	7, 15, 42	syl2anc 411	. . . . . 6 |- ( ph -> ( B + C ) e. QQ )
44		modqcyc2 10685	. . . . . 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 1275	. . . . 5 |- ( ph -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) )
46	41, 45	eqeq12d 2246	. . . 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 ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   + caddc 8095   x. cmul 8097   < clt 8273   - cmin 8409   / cdiv 8911   ZZcz 9540   QQcq 9914   |_cfl 10591   mod cmo 10647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-n0 9462  df-z 9541  df-q 9915  df-rp 9950  df-fl 10593  df-mod 10648

This theorem is referenced by:  modqaddabs  10687  modqaddmod  10688  modqadd12d  10705  modqaddmulmod  10716  moddvds  12440  modsubi  13072  lgsvalmod  15838  lgsmod  15845  lgsne0  15857  lgseisen  15893