Theorem 3dvdsdec 11307

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 8979	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 8978	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2099	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 5700	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 8608	. . . . . . 7 |- 9 e. CC
6		ax-1cn 7535	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 8783	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 7596	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mulid2i 7588	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 5701	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2119	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 5700	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 7590	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 8783	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 7593	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2119	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 3875	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 8877	. . 3 |- 3 e. ZZ
21	7	nn0zi 8870	. . . 4 |- A e. ZZ
22	15	nn0zi 8870	. . . 4 |- B e. ZZ
23		zaddcl 8888	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 418	. . 3 |- ( A + B ) e. ZZ
25		9nn 8682	. . . . . 6 |- 9 e. NN
26	25	nnzi 8869	. . . . 5 |- 9 e. ZZ
27		zmulcl 8901	. . . . 5 |- ( ( 9 e. ZZ /\ A e. ZZ ) -> ( 9 x. A ) e. ZZ )
28	26, 21, 27	mp2an 418	. . . 4 |- ( 9 x. A ) e. ZZ
29		zmulcl 8901	. . . . . . 7 |- ( ( 3 e. ZZ /\ A e. ZZ ) -> ( 3 x. A ) e. ZZ )
30	20, 21, 29	mp2an 418	. . . . . 6 |- ( 3 x. A ) e. ZZ
31		dvdsmul1 11260	. . . . . 6 |- ( ( 3 e. ZZ /\ ( 3 x. A ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. A ) ) )
32	20, 30, 31	mp2an 418	. . . . 5 |- 3 || ( 3 x. ( 3 x. A ) )
33		3t3e9 8671	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2099	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 5700	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 8595	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 7594	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2115	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 3892	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 267	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 11285	. . 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 1280	. 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 1445   class class class wbr 3867  (class class class)co 5690   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   3c3 8572   9c9 8578   NN0cn0 8771   ZZcz 8848  ;cdc 8976   || cdvds 11238

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-5 8582  df-6 8583  df-7 8584  df-8 8585  df-9 8586  df-n0 8772  df-z 8849  df-dec 8977  df-dvds 11239

This theorem is referenced by: (None)