Theorem 3dvds2dec 12048

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 10823	. . . 4 |- ;; A B C = ( ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) + C )
4		sq10e99m1 10822	. . . . . . . 8 |- (; 1 0 ^ 2 ) = (; 9 9 + 1 )
5	4	oveq1i 5935	. . . . . . 7 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 + 1 ) x. A )
6		9nn0 9290	. . . . . . . . . 10 |- 9 e. NN0
7	6, 6	deccl 9488	. . . . . . . . 9 |- ; 9 9 e. NN0
8	7	nn0cni 9278	. . . . . . . 8 |- ; 9 9 e. CC
9		ax-1cn 7989	. . . . . . . 8 |- 1 e. CC
10	1	nn0cni 9278	. . . . . . . 8 |- A e. CC
11	8, 9, 10	adddiri 8054	. . . . . . 7 |- ( (; 9 9 + 1 ) x. A ) = ( (; 9 9 x. A ) + ( 1 x. A ) )
12	10	mullidi 8046	. . . . . . . 8 |- ( 1 x. A ) = A
13	12	oveq2i 5936	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 1 x. A ) ) = ( (; 9 9 x. A ) + A )
14	5, 11, 13	3eqtri 2221	. . . . . 6 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 x. A ) + A )
15		9p1e10 9476	. . . . . . . . 9 |- ( 9 + 1 ) = ; 1 0
16	15	eqcomi 2200	. . . . . . . 8 |- ; 1 0 = ( 9 + 1 )
17	16	oveq1i 5935	. . . . . . 7 |- (; 1 0 x. B ) = ( ( 9 + 1 ) x. B )
18		9cn 9095	. . . . . . . 8 |- 9 e. CC
19	2	nn0cni 9278	. . . . . . . 8 |- B e. CC
20	18, 9, 19	adddiri 8054	. . . . . . 7 |- ( ( 9 + 1 ) x. B ) = ( ( 9 x. B ) + ( 1 x. B ) )
21	19	mullidi 8046	. . . . . . . 8 |- ( 1 x. B ) = B
22	21	oveq2i 5936	. . . . . . 7 |- ( ( 9 x. B ) + ( 1 x. B ) ) = ( ( 9 x. B ) + B )
23	17, 20, 22	3eqtri 2221	. . . . . 6 |- (; 1 0 x. B ) = ( ( 9 x. B ) + B )
24	14, 23	oveq12i 5937	. . . . 5 |- ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) = ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) )
25	24	oveq1i 5935	. . . 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 8048	. . . . . 6 |- (; 9 9 x. A ) e. CC
27	18, 19	mulcli 8048	. . . . . 6 |- ( 9 x. B ) e. CC
28		add4 8204	. . . . . . 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 5940	. . . . . 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 8047	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) e. CC
32	10, 19	addcli 8047	. . . . . 6 |- ( A + B ) e. CC
33		3dvds2dec.c	. . . . . . 7 |- C e. NN0
34	33	nn0cni 9278	. . . . . 6 |- C e. CC
35	31, 32, 34	addassi 8051	. . . . 5 |- ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) )
36		9t11e99 9603	. . . . . . . . . . 11 |- ( 9 x. ; 1 1 ) = ; 9 9
37	36	eqcomi 2200	. . . . . . . . . 10 |- ; 9 9 = ( 9 x. ; 1 1 )
38	37	oveq1i 5935	. . . . . . . . 9 |- (; 9 9 x. A ) = ( ( 9 x. ; 1 1 ) x. A )
39		1nn0 9282	. . . . . . . . . . . 12 |- 1 e. NN0
40	39, 39	deccl 9488	. . . . . . . . . . 11 |- ; 1 1 e. NN0
41	40	nn0cni 9278	. . . . . . . . . 10 |- ; 1 1 e. CC
42	18, 41, 10	mulassi 8052	. . . . . . . . 9 |- ( ( 9 x. ; 1 1 ) x. A ) = ( 9 x. (; 1 1 x. A ) )
43	38, 42	eqtri 2217	. . . . . . . 8 |- (; 9 9 x. A ) = ( 9 x. (; 1 1 x. A ) )
44	43	oveq1i 5935	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
45	41, 10	mulcli 8048	. . . . . . . . 9 |- (; 1 1 x. A ) e. CC
46	18, 45, 19	adddii 8053	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
47	46	eqcomi 2200	. . . . . . 7 |- ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) ) = ( 9 x. ( (; 1 1 x. A ) + B ) )
48		3t3e9 9165	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
49	48	eqcomi 2200	. . . . . . . . 9 |- 9 = ( 3 x. 3 )
50	49	oveq1i 5935	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) )
51		3cn 9082	. . . . . . . . 9 |- 3 e. CC
52	45, 19	addcli 8047	. . . . . . . . 9 |- ( (; 1 1 x. A ) + B ) e. CC
53	51, 51, 52	mulassi 8052	. . . . . . . 8 |- ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
54	50, 53	eqtri 2217	. . . . . . 7 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
55	44, 47, 54	3eqtri 2221	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
56	55	oveq1i 5935	. . . . 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 2221	. . . 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 2221	. . 3 |- ;; A B C = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
59	58	breq2i 4042	. 2 |- ( 3 || ;; A B C <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
60		3z 9372	. . 3 |- 3 e. ZZ
61	1	nn0zi 9365	. . . . 5 |- A e. ZZ
62	2	nn0zi 9365	. . . . 5 |- B e. ZZ
63		zaddcl 9383	. . . . 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 9365	. . . 4 |- C e. ZZ
66		zaddcl 9383	. . . 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 9365	. . . . . . . 8 |- ; 1 1 e. ZZ
69		zmulcl 9396	. . . . . . . 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 9383	. . . . . . 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 9396	. . . . . 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 9396	. . . . 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 11995	. . . . 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 12022	. . 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 1348	. 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 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   2c2 9058   3c3 9059   9c9 9065   NN0cn0 9266   ZZcz 9343  ;cdc 9474   ^cexp 10647   || cdvds 11969

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-seqfrec 10557  df-exp 10648  df-dvds 11970

This theorem is referenced by: (None)