Theorem 3dvds2dec 11873

Description: A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A, B and C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A, B and C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)

Hypotheses

Ref	Expression
3dvdsdec.a	|- A e. NN0
3dvdsdec.b	|- B e. NN0
3dvds2dec.c	|- C e. NN0

Assertion

Ref	Expression
3dvds2dec	|- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )

Proof of Theorem 3dvds2dec

Step	Hyp	Ref	Expression
1		3dvdsdec.a	. . . . 5 |- A e. NN0
2		3dvdsdec.b	. . . . 5 |- B e. NN0
3	1, 2	3dec 10696	. . . 4 |- ;; A B C = ( ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) + C )
4		sq10e99m1 10695	. . . . . . . 8 |- (; 1 0 ^ 2 ) = (; 9 9 + 1 )
5	4	oveq1i 5887	. . . . . . 7 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 + 1 ) x. A )
6		9nn0 9202	. . . . . . . . . 10 |- 9 e. NN0
7	6, 6	deccl 9400	. . . . . . . . 9 |- ; 9 9 e. NN0
8	7	nn0cni 9190	. . . . . . . 8 |- ; 9 9 e. CC
9		ax-1cn 7906	. . . . . . . 8 |- 1 e. CC
10	1	nn0cni 9190	. . . . . . . 8 |- A e. CC
11	8, 9, 10	adddiri 7970	. . . . . . 7 |- ( (; 9 9 + 1 ) x. A ) = ( (; 9 9 x. A ) + ( 1 x. A ) )
12	10	mullidi 7962	. . . . . . . 8 |- ( 1 x. A ) = A
13	12	oveq2i 5888	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 1 x. A ) ) = ( (; 9 9 x. A ) + A )
14	5, 11, 13	3eqtri 2202	. . . . . 6 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 x. A ) + A )
15		9p1e10 9388	. . . . . . . . 9 |- ( 9 + 1 ) = ; 1 0
16	15	eqcomi 2181	. . . . . . . 8 |- ; 1 0 = ( 9 + 1 )
17	16	oveq1i 5887	. . . . . . 7 |- (; 1 0 x. B ) = ( ( 9 + 1 ) x. B )
18		9cn 9009	. . . . . . . 8 |- 9 e. CC
19	2	nn0cni 9190	. . . . . . . 8 |- B e. CC
20	18, 9, 19	adddiri 7970	. . . . . . 7 |- ( ( 9 + 1 ) x. B ) = ( ( 9 x. B ) + ( 1 x. B ) )
21	19	mullidi 7962	. . . . . . . 8 |- ( 1 x. B ) = B
22	21	oveq2i 5888	. . . . . . 7 |- ( ( 9 x. B ) + ( 1 x. B ) ) = ( ( 9 x. B ) + B )
23	17, 20, 22	3eqtri 2202	. . . . . 6 |- (; 1 0 x. B ) = ( ( 9 x. B ) + B )
24	14, 23	oveq12i 5889	. . . . 5 |- ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) = ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) )
25	24	oveq1i 5887	. . . 4 |- ( ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) + C ) = ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C )
26	8, 10	mulcli 7964	. . . . . 6 |- (; 9 9 x. A ) e. CC
27	18, 19	mulcli 7964	. . . . . 6 |- ( 9 x. B ) e. CC
28		add4 8120	. . . . . . 7 |- ( ( ( (; 9 9 x. A ) e. CC /\ A e. CC ) /\ ( ( 9 x. B ) e. CC /\ B e. CC ) ) -> ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) )
29	28	oveq1d 5892	. . . . . 6 |- ( ( ( (; 9 9 x. A ) e. CC /\ A e. CC ) /\ ( ( 9 x. B ) e. CC /\ B e. CC ) ) -> ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) )
30	26, 10, 27, 19, 29	mp4an 427	. . . . 5 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C )
31	26, 27	addcli 7963	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) e. CC
32	10, 19	addcli 7963	. . . . . 6 |- ( A + B ) e. CC
33		3dvds2dec.c	. . . . . . 7 |- C e. NN0
34	33	nn0cni 9190	. . . . . 6 |- C e. CC
35	31, 32, 34	addassi 7967	. . . . 5 |- ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) )
36		9t11e99 9515	. . . . . . . . . . 11 |- ( 9 x. ; 1 1 ) = ; 9 9
37	36	eqcomi 2181	. . . . . . . . . 10 |- ; 9 9 = ( 9 x. ; 1 1 )
38	37	oveq1i 5887	. . . . . . . . 9 |- (; 9 9 x. A ) = ( ( 9 x. ; 1 1 ) x. A )
39		1nn0 9194	. . . . . . . . . . . 12 |- 1 e. NN0
40	39, 39	deccl 9400	. . . . . . . . . . 11 |- ; 1 1 e. NN0
41	40	nn0cni 9190	. . . . . . . . . 10 |- ; 1 1 e. CC
42	18, 41, 10	mulassi 7968	. . . . . . . . 9 |- ( ( 9 x. ; 1 1 ) x. A ) = ( 9 x. (; 1 1 x. A ) )
43	38, 42	eqtri 2198	. . . . . . . 8 |- (; 9 9 x. A ) = ( 9 x. (; 1 1 x. A ) )
44	43	oveq1i 5887	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
45	41, 10	mulcli 7964	. . . . . . . . 9 |- (; 1 1 x. A ) e. CC
46	18, 45, 19	adddii 7969	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
47	46	eqcomi 2181	. . . . . . 7 |- ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) ) = ( 9 x. ( (; 1 1 x. A ) + B ) )
48		3t3e9 9078	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
49	48	eqcomi 2181	. . . . . . . . 9 |- 9 = ( 3 x. 3 )
50	49	oveq1i 5887	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) )
51		3cn 8996	. . . . . . . . 9 |- 3 e. CC
52	45, 19	addcli 7963	. . . . . . . . 9 |- ( (; 1 1 x. A ) + B ) e. CC
53	51, 51, 52	mulassi 7968	. . . . . . . 8 |- ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
54	50, 53	eqtri 2198	. . . . . . 7 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
55	44, 47, 54	3eqtri 2202	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
56	55	oveq1i 5887	. . . . 5 |- ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
57	30, 35, 56	3eqtri 2202	. . . 4 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
58	3, 25, 57	3eqtri 2202	. . 3 |- ;; A B C = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
59	58	breq2i 4013	. 2 |- ( 3 || ;; A B C <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
60		3z 9284	. . 3 |- 3 e. ZZ
61	1	nn0zi 9277	. . . . 5 |- A e. ZZ
62	2	nn0zi 9277	. . . . 5 |- B e. ZZ
63		zaddcl 9295	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
64	61, 62, 63	mp2an 426	. . . 4 |- ( A + B ) e. ZZ
65	33	nn0zi 9277	. . . 4 |- C e. ZZ
66		zaddcl 9295	. . . 4 |- ( ( ( A + B ) e. ZZ /\ C e. ZZ ) -> ( ( A + B ) + C ) e. ZZ )
67	64, 65, 66	mp2an 426	. . 3 |- ( ( A + B ) + C ) e. ZZ
68	40	nn0zi 9277	. . . . . . . 8 |- ; 1 1 e. ZZ
69		zmulcl 9308	. . . . . . . 8 |- ( (; 1 1 e. ZZ /\ A e. ZZ ) -> (; 1 1 x. A ) e. ZZ )
70	68, 61, 69	mp2an 426	. . . . . . 7 |- (; 1 1 x. A ) e. ZZ
71		zaddcl 9295	. . . . . . 7 |- ( ( (; 1 1 x. A ) e. ZZ /\ B e. ZZ ) -> ( (; 1 1 x. A ) + B ) e. ZZ )
72	70, 62, 71	mp2an 426	. . . . . 6 |- ( (; 1 1 x. A ) + B ) e. ZZ
73		zmulcl 9308	. . . . . 6 |- ( ( 3 e. ZZ /\ ( (; 1 1 x. A ) + B ) e. ZZ ) -> ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ )
74	60, 72, 73	mp2an 426	. . . . 5 |- ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ
75		zmulcl 9308	. . . . 5 |- ( ( 3 e. ZZ /\ ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ ) -> ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ )
76	60, 74, 75	mp2an 426	. . . 4 |- ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ
77		dvdsmul1 11822	. . . . 5 |- ( ( 3 e. ZZ /\ ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) )
78	60, 74, 77	mp2an 426	. . . 4 |- 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
79	76, 78	pm3.2i 272	. . 3 |- ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ /\ 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) )
80		dvdsadd2b 11849	. . 3 |- ( ( 3 e. ZZ /\ ( ( A + B ) + C ) e. ZZ /\ ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ /\ 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) ) ) -> ( 3 || ( ( A + B ) + C ) <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) ) )
81	60, 67, 79, 80	mp3an 1337	. 2 |- ( 3 || ( ( A + B ) + C ) <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
82	59, 81	bitr4i 187	1 |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   2c2 8972   3c3 8973   9c9 8979   NN0cn0 9178   ZZcz 9255  ;cdc 9386   ^cexp 10521   || cdvds 11796

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-n0 9179  df-z 9256  df-dec 9387  df-uz 9531  df-seqfrec 10448  df-exp 10522  df-dvds 11797

This theorem is referenced by: (None)