Theorem 3dvdsdec 12342

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 9549	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
2		9p1e10 9548	. . . . . . . 8 ⊢ (9 + 1) = ;10
3	2	eqcomi 2213	. . . . . . 7 ⊢ ;10 = (9 + 1)
4	3	oveq1i 5984	. . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴)
5		9cn 9166	. . . . . . 7 ⊢ 9 ∈ ℂ
6		ax-1cn 8060	. . . . . . 7 ⊢ 1 ∈ ℂ
7		3dvdsdec.a	. . . . . . . 8 ⊢ 𝐴 ∈ ℕ0
8	7	nn0cni 9349	. . . . . . 7 ⊢ 𝐴 ∈ ℂ
9	5, 6, 8	adddiri 8125	. . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴))
10	8	mullidi 8117	. . . . . . 7 ⊢ (1 · 𝐴) = 𝐴
11	10	oveq2i 5985	. . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴)
12	4, 9, 11	3eqtri 2234	. . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴)
13	12	oveq1i 5984	. . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵)
14	5, 8	mulcli 8119	. . . . 5 ⊢ (9 · 𝐴) ∈ ℂ
15		3dvdsdec.b	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
16	15	nn0cni 9349	. . . . 5 ⊢ 𝐵 ∈ ℂ
17	14, 8, 16	addassi 8122	. . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵))
18	1, 13, 17	3eqtri 2234	. . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵))
19	18	breq2i 4070	. 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵)))
20		3z 9443	. . 3 ⊢ 3 ∈ ℤ
21	7	nn0zi 9436	. . . 4 ⊢ 𝐴 ∈ ℤ
22	15	nn0zi 9436	. . . 4 ⊢ 𝐵 ∈ ℤ
23		zaddcl 9454	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ)
24	21, 22, 23	mp2an 426	. . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ
25		9nn 9247	. . . . . 6 ⊢ 9 ∈ ℕ
26	25	nnzi 9435	. . . . 5 ⊢ 9 ∈ ℤ
27		zmulcl 9468	. . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ)
28	26, 21, 27	mp2an 426	. . . 4 ⊢ (9 · 𝐴) ∈ ℤ
29		zmulcl 9468	. . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ)
30	20, 21, 29	mp2an 426	. . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ
31		dvdsmul1 12290	. . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴)))
32	20, 30, 31	mp2an 426	. . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴))
33		3t3e9 9236	. . . . . . . 8 ⊢ (3 · 3) = 9
34	33	eqcomi 2213	. . . . . . 7 ⊢ 9 = (3 · 3)
35	34	oveq1i 5984	. . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴)
36		3cn 9153	. . . . . . 7 ⊢ 3 ∈ ℂ
37	36, 36, 8	mulassi 8123	. . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴))
38	35, 37	eqtri 2230	. . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴))
39	32, 38	breqtrri 4089	. . . 4 ⊢ 3 ∥ (9 · 𝐴)
40	28, 39	pm3.2i 272	. . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴))
41		dvdsadd2b 12317	. . 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 1352	. 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵)))
43	19, 42	bitr4i 187	1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2180   class class class wbr 4062  (class class class)co 5974  0cc0 7967  1c1 7968   + caddc 7970   · cmul 7972  3c3 9130  9c9 9136  ℕ₀cn0 9337  ℤcz 9414  ;cdc 9546   ∥ cdvds 12264

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-dvds 12265

This theorem is referenced by: (None)