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Mirrors > Home > ILE Home > Th. List > 3dvdsdec | GIF version |
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.) |
Ref | Expression |
---|---|
3dvdsdec.a | ⊢ 𝐴 ∈ ℕ0 |
3dvdsdec.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
3dvdsdec | ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 9082 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 9p1e10 9081 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
3 | 2 | eqcomi 2117 | . . . . . . 7 ⊢ ;10 = (9 + 1) |
4 | 3 | oveq1i 5736 | . . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴) |
5 | 9cn 8711 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
6 | ax-1cn 7631 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
7 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 8886 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | 5, 6, 8 | adddiri 7694 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
10 | 8 | mulid2i 7686 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
11 | 10 | oveq2i 5737 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
12 | 4, 9, 11 | 3eqtri 2137 | . . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
13 | 12 | oveq1i 5736 | . . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
14 | 5, 8 | mulcli 7688 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
15 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
16 | 15 | nn0cni 8886 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
17 | 14, 8, 16 | addassi 7691 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
18 | 1, 13, 17 | 3eqtri 2137 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
19 | 18 | breq2i 3901 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
20 | 3z 8980 | . . 3 ⊢ 3 ∈ ℤ | |
21 | 7 | nn0zi 8973 | . . . 4 ⊢ 𝐴 ∈ ℤ |
22 | 15 | nn0zi 8973 | . . . 4 ⊢ 𝐵 ∈ ℤ |
23 | zaddcl 8991 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
24 | 21, 22, 23 | mp2an 420 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
25 | 9nn 8785 | . . . . . 6 ⊢ 9 ∈ ℕ | |
26 | 25 | nnzi 8972 | . . . . 5 ⊢ 9 ∈ ℤ |
27 | zmulcl 9004 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
28 | 26, 21, 27 | mp2an 420 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
29 | zmulcl 9004 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
30 | 20, 21, 29 | mp2an 420 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
31 | dvdsmul1 11356 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
32 | 20, 30, 31 | mp2an 420 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
33 | 3t3e9 8774 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2117 | . . . . . . 7 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 5736 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
36 | 3cn 8698 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
37 | 36, 36, 8 | mulassi 7692 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
38 | 35, 37 | eqtri 2133 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
39 | 32, 38 | breqtrri 3918 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
40 | 28, 39 | pm3.2i 268 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
41 | dvdsadd2b 11381 | . . 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 1296 | . 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
43 | 19, 42 | bitr4i 186 | 1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1461 class class class wbr 3893 (class class class)co 5726 0cc0 7540 1c1 7541 + caddc 7543 · cmul 7545 3c3 8675 9c9 8681 ℕ0cn0 8874 ℤcz 8951 ;cdc 9079 ∥ cdvds 11334 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7629 ax-resscn 7630 ax-1cn 7631 ax-1re 7632 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-mulrcl 7637 ax-addcom 7638 ax-mulcom 7639 ax-addass 7640 ax-mulass 7641 ax-distr 7642 ax-i2m1 7643 ax-0lt1 7644 ax-1rid 7645 ax-0id 7646 ax-rnegex 7647 ax-cnre 7649 ax-pre-ltirr 7650 ax-pre-ltwlin 7651 ax-pre-lttrn 7652 ax-pre-ltadd 7654 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7719 df-mnf 7720 df-xr 7721 df-ltxr 7722 df-le 7723 df-sub 7851 df-neg 7852 df-inn 8624 df-2 8682 df-3 8683 df-4 8684 df-5 8685 df-6 8686 df-7 8687 df-8 8688 df-9 8689 df-n0 8875 df-z 8952 df-dec 9080 df-dvds 11335 |
This theorem is referenced by: (None) |
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