Theorem 3dvdsdec 12416

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 9604	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
2		9p1e10 9603	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
3	2	eqcomi 2233	. . . . . . 7 |- ; 1 0 = ( 9 + 1 )
4	3	oveq1i 6023	. . . . . 6 |- (; 1 0 x. A ) = ( ( 9 + 1 ) x. A )
5		9cn 9221	. . . . . . 7 |- 9 e. CC
6		ax-1cn 8115	. . . . . . 7 |- 1 e. CC
7		3dvdsdec.a	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 9404	. . . . . . 7 |- A e. CC
9	5, 6, 8	adddiri 8180	. . . . . 6 |- ( ( 9 + 1 ) x. A ) = ( ( 9 x. A ) + ( 1 x. A ) )
10	8	mullidi 8172	. . . . . . 7 |- ( 1 x. A ) = A
11	10	oveq2i 6024	. . . . . 6 |- ( ( 9 x. A ) + ( 1 x. A ) ) = ( ( 9 x. A ) + A )
12	4, 9, 11	3eqtri 2254	. . . . 5 |- (; 1 0 x. A ) = ( ( 9 x. A ) + A )
13	12	oveq1i 6023	. . . 4 |- ( (; 1 0 x. A ) + B ) = ( ( ( 9 x. A ) + A ) + B )
14	5, 8	mulcli 8174	. . . . 5 |- ( 9 x. A ) e. CC
15		3dvdsdec.b	. . . . . 6 |- B e. NN0
16	15	nn0cni 9404	. . . . 5 |- B e. CC
17	14, 8, 16	addassi 8177	. . . 4 |- ( ( ( 9 x. A ) + A ) + B ) = ( ( 9 x. A ) + ( A + B ) )
18	1, 13, 17	3eqtri 2254	. . 3 |- ; A B = ( ( 9 x. A ) + ( A + B ) )
19	18	breq2i 4094	. 2 |- ( 3 || ; A B <-> 3 || ( ( 9 x. A ) + ( A + B ) ) )
20		3z 9498	. . 3 |- 3 e. ZZ
21	7	nn0zi 9491	. . . 4 |- A e. ZZ
22	15	nn0zi 9491	. . . 4 |- B e. ZZ
23		zaddcl 9509	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
24	21, 22, 23	mp2an 426	. . 3 |- ( A + B ) e. ZZ
25		9nn 9302	. . . . . 6 |- 9 e. NN
26	25	nnzi 9490	. . . . 5 |- 9 e. ZZ
27		zmulcl 9523	. . . . 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 9523	. . . . . . 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 12364	. . . . . 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 9291	. . . . . . . 8 |- ( 3 x. 3 ) = 9
34	33	eqcomi 2233	. . . . . . 7 |- 9 = ( 3 x. 3 )
35	34	oveq1i 6023	. . . . . 6 |- ( 9 x. A ) = ( ( 3 x. 3 ) x. A )
36		3cn 9208	. . . . . . 7 |- 3 e. CC
37	36, 36, 8	mulassi 8178	. . . . . 6 |- ( ( 3 x. 3 ) x. A ) = ( 3 x. ( 3 x. A ) )
38	35, 37	eqtri 2250	. . . . 5 |- ( 9 x. A ) = ( 3 x. ( 3 x. A ) )
39	32, 38	breqtrri 4113	. . . 4 |- 3 || ( 9 x. A )
40	28, 39	pm3.2i 272	. . 3 |- ( ( 9 x. A ) e. ZZ /\ 3 || ( 9 x. A ) )
41		dvdsadd2b 12391	. . 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 1371	. 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 2200   class class class wbr 4086  (class class class)co 6013   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   3c3 9185   9c9 9191   NN0cn0 9392   ZZcz 9469  ;cdc 9601   || cdvds 12338

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-dec 9602  df-dvds 12339

This theorem is referenced by: (None)