Theorem 3dvdsdec 11562

Description: A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A and B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A and B, especially if A is itself a decimal number, e.g. A = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)

Hypotheses

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

Assertion

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

Proof of Theorem 3dvdsdec

Step	Hyp	Ref	Expression
1		dfdec10 9185	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 9184	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2143	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 5784	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 8808	. . . . . . 7 |- 9 e. CC
6		ax-1cn 7713	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 8989	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 7777	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mulid2i 7769	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 5785	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2164	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 5784	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 7771	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 8989	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 7774	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2164	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 3937	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 9083	. . 3 |- 3 e. ZZ
21	7	nn0zi 9076	. . . 4 |- A e. ZZ
22	15	nn0zi 9076	. . . 4 |- B e. ZZ
23		zaddcl 9094	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 422	. . 3 |- ( A + B ) e. ZZ
25		9nn 8888	. . . . . 6 |- 9 e. NN
26	25	nnzi 9075	. . . . 5 |- 9 e. ZZ
27		zmulcl 9107	. . . . 5 |- ( ( 9 e. ZZ /\ A e. ZZ ) -> ( 9 x. A ) e. ZZ )
28	26, 21, 27	mp2an 422	. . . 4 |- ( 9 x. A ) e. ZZ
29		zmulcl 9107	. . . . . . 7 |- ( ( 3 e. ZZ /\ A e. ZZ ) -> ( 3 x. A ) e. ZZ )
30	20, 21, 29	mp2an 422	. . . . . 6 |- ( 3 x. A ) e. ZZ
31		dvdsmul1 11515	. . . . . 6 |- ( ( 3 e. ZZ /\ ( 3 x. A ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. A ) ) )
32	20, 30, 31	mp2an 422	. . . . 5 |- 3 || ( 3 x. ( 3 x. A ) )
33		3t3e9 8877	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2143	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 5784	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 8795	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 7775	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2160	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 3955	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 270	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 11540	. . 3 |- ( ( 3 e. ZZ /\ ( A + B ) e. ZZ /\ ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) ) ) -> ( 3 || ( A + B ) <-> 3 || ( ( 9 x. A ) + ( A + B ) ) ) )
42	20, 24, 40, 41	mp3an 1315	. 2 |- ( 3 || ( A + B ) <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
43	19, 42	bitr4i 186	1 |- ( 3 || ; A B <-> 3 || ( A + B ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   3c3 8772   9c9 8778   NN0cn0 8977   ZZcz 9054  ;cdc 9182   || cdvds 11493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-7 8784  df-8 8785  df-9 8786  df-n0 8978  df-z 9055  df-dec 9183  df-dvds 11494

This theorem is referenced by: (None)