Theorem 3dvds2dec 11854

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 10678	. . . 4 |- ;; A B C = ( ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) + C )
4		sq10e99m1 10677	. . . . . . . 8 |- (; 1 0 ^ 2 ) = (; 9 9 + 1 )
5	4	oveq1i 5879	. . . . . . 7 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 + 1 ) x. A )
6		9nn0 9189	. . . . . . . . . 10 |- 9 e. NN0
7	6, 6	deccl 9387	. . . . . . . . 9 |- ; 9 9 e. NN0
8	7	nn0cni 9177	. . . . . . . 8 |- ; 9 9 e. CC
9		ax-1cn 7895	. . . . . . . 8 |- 1 e. CC
10	1	nn0cni 9177	. . . . . . . 8 |- A e. CC
11	8, 9, 10	adddiri 7959	. . . . . . 7 |- ( (; 9 9 + 1 ) x. A ) = ( (; 9 9 x. A ) + ( 1 x. A ) )
12	10	mulid2i 7951	. . . . . . . 8 |- ( 1 x. A ) = A
13	12	oveq2i 5880	. . . . . . 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 9375	. . . . . . . . 9 |- ( 9 + 1 ) = ; 1 0
16	15	eqcomi 2181	. . . . . . . 8 |- ; 1 0 = ( 9 + 1 )
17	16	oveq1i 5879	. . . . . . 7 |- (; 1 0 x. B ) = ( ( 9 + 1 ) x. B )
18		9cn 8996	. . . . . . . 8 |- 9 e. CC
19	2	nn0cni 9177	. . . . . . . 8 |- B e. CC
20	18, 9, 19	adddiri 7959	. . . . . . 7 |- ( ( 9 + 1 ) x. B ) = ( ( 9 x. B ) + ( 1 x. B ) )
21	19	mulid2i 7951	. . . . . . . 8 |- ( 1 x. B ) = B
22	21	oveq2i 5880	. . . . . . 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 5881	. . . . 5 |- ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) = ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) )
25	24	oveq1i 5879	. . . 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 7953	. . . . . 6 |- (; 9 9 x. A ) e. CC
27	18, 19	mulcli 7953	. . . . . 6 |- ( 9 x. B ) e. CC
28		add4 8108	. . . . . . 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 5884	. . . . . 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 7952	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) e. CC
32	10, 19	addcli 7952	. . . . . 6 |- ( A + B ) e. CC
33		3dvds2dec.c	. . . . . . 7 |- C e. NN0
34	33	nn0cni 9177	. . . . . 6 |- C e. CC
35	31, 32, 34	addassi 7956	. . . . 5 |- ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) )
36		9t11e99 9502	. . . . . . . . . . 11 |- ( 9 x. ; 1 1 ) = ; 9 9
37	36	eqcomi 2181	. . . . . . . . . 10 |- ; 9 9 = ( 9 x. ; 1 1 )
38	37	oveq1i 5879	. . . . . . . . 9 |- (; 9 9 x. A ) = ( ( 9 x. ; 1 1 ) x. A )
39		1nn0 9181	. . . . . . . . . . . 12 |- 1 e. NN0
40	39, 39	deccl 9387	. . . . . . . . . . 11 |- ; 1 1 e. NN0
41	40	nn0cni 9177	. . . . . . . . . 10 |- ; 1 1 e. CC
42	18, 41, 10	mulassi 7957	. . . . . . . . 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 5879	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
45	41, 10	mulcli 7953	. . . . . . . . 9 |- (; 1 1 x. A ) e. CC
46	18, 45, 19	adddii 7958	. . . . . . . 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 9065	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
49	48	eqcomi 2181	. . . . . . . . 9 |- 9 = ( 3 x. 3 )
50	49	oveq1i 5879	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) )
51		3cn 8983	. . . . . . . . 9 |- 3 e. CC
52	45, 19	addcli 7952	. . . . . . . . 9 |- ( (; 1 1 x. A ) + B ) e. CC
53	51, 51, 52	mulassi 7957	. . . . . . . 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 5879	. . . . 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 4008	. 2 |- ( 3 || ;; A B C <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
60		3z 9271	. . 3 |- 3 e. ZZ
61	1	nn0zi 9264	. . . . 5 |- A e. ZZ
62	2	nn0zi 9264	. . . . 5 |- B e. ZZ
63		zaddcl 9282	. . . . 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 9264	. . . 4 |- C e. ZZ
66		zaddcl 9282	. . . 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 9264	. . . . . . . 8 |- ; 1 1 e. ZZ
69		zmulcl 9295	. . . . . . . 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 9282	. . . . . . 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 9295	. . . . . 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 9295	. . . . 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 11804	. . . . 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 11831	. . 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 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   2c2 8959   3c3 8960   9c9 8966   NN0cn0 9165   ZZcz 9242  ;cdc 9373   ^cexp 10505   || cdvds 11778

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-dec 9374  df-uz 9518  df-seqfrec 10432  df-exp 10506  df-dvds 11779

This theorem is referenced by: (None)