Theorem 3dvdsdec 11860

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 9381	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 9380	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2181	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 5880	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 9001	. . . . . . 7 |- 9 e. CC
6		ax-1cn 7899	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 9182	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 7963	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mulid2i 7955	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 5881	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2202	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 5880	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 7957	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 9182	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 7960	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2202	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 4009	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 9276	. . 3 |- 3 e. ZZ
21	7	nn0zi 9269	. . . 4 |- A e. ZZ
22	15	nn0zi 9269	. . . 4 |- B e. ZZ
23		zaddcl 9287	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 426	. . 3 |- ( A + B ) e. ZZ
25		9nn 9081	. . . . . 6 |- 9 e. NN
26	25	nnzi 9268	. . . . 5 |- 9 e. ZZ
27		zmulcl 9300	. . . . 5 |- ( ( 9 e. ZZ /\ A e. ZZ ) -> ( 9 x. A ) e. ZZ )
28	26, 21, 27	mp2an 426	. . . 4 |- ( 9 x. A ) e. ZZ
29		zmulcl 9300	. . . . . . 7 |- ( ( 3 e. ZZ /\ A e. ZZ ) -> ( 3 x. A ) e. ZZ )
30	20, 21, 29	mp2an 426	. . . . . 6 |- ( 3 x. A ) e. ZZ
31		dvdsmul1 11811	. . . . . 6 |- ( ( 3 e. ZZ /\ ( 3 x. A ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. A ) ) )
32	20, 30, 31	mp2an 426	. . . . 5 |- 3 || ( 3 x. ( 3 x. A ) )
33		3t3e9 9070	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2181	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 5880	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 8988	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 7961	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2198	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 4028	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 272	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 11838	. . 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 1337	. 2 |- ( 3 || ( A + B ) <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
43	19, 42	bitr4i 187	1 |- ( 3 || ; A B <-> 3 || ( A + B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2148   class class class wbr 4001  (class class class)co 5870   0cc0 7806   1c1 7807   + caddc 7809   x. cmul 7811   3c3 8965   9c9 8971   NN0cn0 9170   ZZcz 9247  ;cdc 9378   || cdvds 11785

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-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-ltadd 7922

This theorem depends on definitions:  df-bi 117  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-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-br 4002  df-opab 4063  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-iota 5175  df-fun 5215  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-inn 8914  df-2 8972  df-3 8973  df-4 8974  df-5 8975  df-6 8976  df-7 8977  df-8 8978  df-9 8979  df-n0 9171  df-z 9248  df-dec 9379  df-dvds 11786

This theorem is referenced by: (None)