Theorem 3dvdsdec 10790

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 8815	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 8814	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2089	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 5625	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 8448	. . . . . . 7 |- 9 e. CC
6		ax-1cn 7385	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 8621	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 7446	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mulid2i 7438	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 5626	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2109	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 5625	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 7440	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 8621	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 7443	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2109	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 3830	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 8715	. . 3 |- 3 e. ZZ
21	7	nn0zi 8708	. . . 4 |- A e. ZZ
22	15	nn0zi 8708	. . . 4 |- B e. ZZ
23		zaddcl 8726	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 417	. . 3 |- ( A + B ) e. ZZ
25		9nn 8521	. . . . . 6 |- 9 e. NN
26	25	nnzi 8707	. . . . 5 |- 9 e. ZZ
27		zmulcl 8739	. . . . 5 |- ( ( 9 e. ZZ /\ A e. ZZ ) -> ( 9 x. A ) e. ZZ )
28	26, 21, 27	mp2an 417	. . . 4 |- ( 9 x. A ) e. ZZ
29		zmulcl 8739	. . . . . . 7 |- ( ( 3 e. ZZ /\ A e. ZZ ) -> ( 3 x. A ) e. ZZ )
30	20, 21, 29	mp2an 417	. . . . . 6 |- ( 3 x. A ) e. ZZ
31		dvdsmul1 10743	. . . . . 6 |- ( ( 3 e. ZZ /\ ( 3 x. A ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. A ) ) )
32	20, 30, 31	mp2an 417	. . . . 5 |- 3 || ( 3 x. ( 3 x. A ) )
33		3t3e9 8510	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2089	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 5625	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 8435	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 7444	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2105	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 3847	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 266	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 10768	. . 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 1271	. 2 |- ( 3 || ( A + B ) <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
43	19, 42	bitr4i 185	1 |- ( 3 || ; A B <-> 3 || ( A + B ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1436   class class class wbr 3822  (class class class)co 5615   0cc0 7297   1c1 7298   + caddc 7300   x. cmul 7302   3c3 8411   9c9 8417   NN0cn0 8609   ZZcz 8686  ;cdc 8812   || cdvds 10721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-ltadd 7408

This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-inn 8361  df-2 8419  df-3 8420  df-4 8421  df-5 8422  df-6 8423  df-7 8424  df-8 8425  df-9 8426  df-n0 8610  df-z 8687  df-dec 8813  df-dvds 10722

This theorem is referenced by: (None)