Theorem 3dvdsdec 10264

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 = ;𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)

Hypotheses

Ref	Expression
3dvdsdec.a	⊢ 𝐴 ∈ ℕ0
3dvdsdec.b	⊢ 𝐵 ∈ ℕ0

Assertion

Ref	Expression
3dvdsdec	⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Proof of Theorem 3dvdsdec

Step	Hyp	Ref	Expression
1		dfdec10 8480	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
2		9p1e10 8479	. . . . . . . 8 ⊢ (9 + 1) = ;10
3	2	eqcomi 2085	. . . . . . 7 ⊢ ;10 = (9 + 1)
4	3	oveq1i 5542	. . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴)
5		9cn 8127	. . . . . . 7 ⊢ 9 ∈ ℂ
6		ax-1cn 7069	. . . . . . 7 ⊢ 1 ∈ ℂ
7		3dvdsdec.a	. . . . . . . 8 ⊢ 𝐴 ∈ ℕ0
8	7	nn0cni 8300	. . . . . . 7 ⊢ 𝐴 ∈ ℂ
9	5, 6, 8	adddiri 7130	. . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴))
10	8	mulid2i 7122	. . . . . . 7 ⊢ (1 · 𝐴) = 𝐴
11	10	oveq2i 5543	. . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴)
12	4, 9, 11	3eqtri 2105	. . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴)
13	12	oveq1i 5542	. . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵)
14	5, 8	mulcli 7124	. . . . 5 ⊢ (9 · 𝐴) ∈ ℂ
15		3dvdsdec.b	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
16	15	nn0cni 8300	. . . . 5 ⊢ 𝐵 ∈ ℂ
17	14, 8, 16	addassi 7127	. . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵))
18	1, 13, 17	3eqtri 2105	. . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵))
19	18	breq2i 3793	. 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵)))
20		3z 8380	. . 3 ⊢ 3 ∈ ℤ
21	7	nn0zi 8373	. . . 4 ⊢ 𝐴 ∈ ℤ
22	15	nn0zi 8373	. . . 4 ⊢ 𝐵 ∈ ℤ
23		zaddcl 8391	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ)
24	21, 22, 23	mp2an 416	. . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ
25		9nn 8200	. . . . . 6 ⊢ 9 ∈ ℕ
26	25	nnzi 8372	. . . . 5 ⊢ 9 ∈ ℤ
27		zmulcl 8404	. . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ)
28	26, 21, 27	mp2an 416	. . . 4 ⊢ (9 · 𝐴) ∈ ℤ
29		zmulcl 8404	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ)
30	20, 21, 29	mp2an 416	. . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ
31		dvdsmul1 10217	. . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴)))
32	20, 30, 31	mp2an 416	. . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴))
33		3t3e9 8189	. . . . . . . 8 ⊢ (3 · 3) = 9
34	33	eqcomi 2085	. . . . . . 7 ⊢ 9 = (3 · 3)
35	34	oveq1i 5542	. . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴)
36		3cn 8114	. . . . . . 7 ⊢ 3 ∈ ℂ
37	36, 36, 8	mulassi 7128	. . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴))
38	35, 37	eqtri 2101	. . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴))
39	32, 38	breqtrri 3810	. . . 4 ⊢ 3 ∥ (9 · 𝐴)
40	28, 39	pm3.2i 266	. . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴))
41		dvdsadd2b 10242	. . 3 ⊢ ((3 ∈ ℤ ∧ (𝐴 + 𝐵) ∈ ℤ ∧ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴))) → (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))))
42	20, 24, 40, 41	mp3an 1268	. 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵)))
43	19, 42	bitr4i 185	1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Syntax hints:   ∧ wa 102   ↔ wb 103   ∈ wcel 1433   class class class wbr 3785  (class class class)co 5532  0cc0 6981  1c1 6982   + caddc 6984   · cmul 6986  3c3 8090  9c9 8096  ℕ₀cn0 8288  ℤcz 8351  ;cdc 8477   ∥ cdvds 10195

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-5 8101  df-6 8102  df-7 8103  df-8 8104  df-9 8105  df-n0 8289  df-z 8352  df-dec 8478  df-dvds 10196

This theorem is referenced by: (None)