Theorem 3dvdsdec 12009

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 9454	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 9453	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2197	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 5929	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 9072	. . . . . . 7 |- 9 e. CC
6		ax-1cn 7967	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 9255	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 8032	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mullidi 8024	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 5930	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2218	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 5929	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 8026	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 9255	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 8029	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2218	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 4038	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 9349	. . 3 |- 3 e. ZZ
21	7	nn0zi 9342	. . . 4 |- A e. ZZ
22	15	nn0zi 9342	. . . 4 |- B e. ZZ
23		zaddcl 9360	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 426	. . 3 |- ( A + B ) e. ZZ
25		9nn 9153	. . . . . 6 |- 9 e. NN
26	25	nnzi 9341	. . . . 5 |- 9 e. ZZ
27		zmulcl 9373	. . . . 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 9373	. . . . . . 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 11959	. . . . . 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 9142	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2197	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 5929	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 9059	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 8030	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2214	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 4057	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 272	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 11986	. . 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 1348	. 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 2164   class class class wbr 4030  (class class class)co 5919   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   3c3 9036   9c9 9042   NN0cn0 9243   ZZcz 9320  ;cdc 9451   || cdvds 11933

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-dec 9452  df-dvds 11934

This theorem is referenced by: (None)