Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9316 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 9p1e10 9315 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
3 | 2 | eqcomi 2168 | . . . . . . 7 ⊢ ;10 = (9 + 1) |
4 | 3 | oveq1i 5846 | . . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴) |
5 | 9cn 8936 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
6 | ax-1cn 7837 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
7 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 9117 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | 5, 6, 8 | adddiri 7901 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
10 | 8 | mulid2i 7893 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
11 | 10 | oveq2i 5847 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
12 | 4, 9, 11 | 3eqtri 2189 | . . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
13 | 12 | oveq1i 5846 | . . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
14 | 5, 8 | mulcli 7895 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
15 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
16 | 15 | nn0cni 9117 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
17 | 14, 8, 16 | addassi 7898 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
18 | 1, 13, 17 | 3eqtri 2189 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
19 | 18 | breq2i 3984 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
20 | 3z 9211 | . . 3 ⊢ 3 ∈ ℤ | |
21 | 7 | nn0zi 9204 | . . . 4 ⊢ 𝐴 ∈ ℤ |
22 | 15 | nn0zi 9204 | . . . 4 ⊢ 𝐵 ∈ ℤ |
23 | zaddcl 9222 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
24 | 21, 22, 23 | mp2an 423 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
25 | 9nn 9016 | . . . . . 6 ⊢ 9 ∈ ℕ | |
26 | 25 | nnzi 9203 | . . . . 5 ⊢ 9 ∈ ℤ |
27 | zmulcl 9235 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
28 | 26, 21, 27 | mp2an 423 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
29 | zmulcl 9235 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
30 | 20, 21, 29 | mp2an 423 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
31 | dvdsmul1 11739 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
32 | 20, 30, 31 | mp2an 423 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
33 | 3t3e9 9005 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2168 | . . . . . . 7 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 5846 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
36 | 3cn 8923 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
37 | 36, 36, 8 | mulassi 7899 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
38 | 35, 37 | eqtri 2185 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
39 | 32, 38 | breqtrri 4003 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
40 | 28, 39 | pm3.2i 270 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
41 | dvdsadd2b 11765 | . . 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 1326 | . 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 2135 class class class wbr 3976 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 3c3 8900 9c9 8906 ℕ0cn0 9105 ℤcz 9182 ;cdc 9313 ∥ cdvds 11713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-dec 9314 df-dvds 11714 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |