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Mirrors > Home > MPE Home > Th. List > 3dvdsdec | Structured version Visualization version 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 12261 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 9p1e10 12260 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
3 | 2 | eqcomi 2745 | . . . . . . 7 ⊢ ;10 = (9 + 1) |
4 | 3 | oveq1i 7201 | . . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴) |
5 | 9cn 11895 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
6 | ax-1cn 10752 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
7 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 12067 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | 5, 6, 8 | adddiri 10811 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
10 | 8 | mulid2i 10803 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
11 | 10 | oveq2i 7202 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
12 | 4, 9, 11 | 3eqtri 2763 | . . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
13 | 12 | oveq1i 7201 | . . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
14 | 5, 8 | mulcli 10805 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
15 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
16 | 15 | nn0cni 12067 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
17 | 14, 8, 16 | addassi 10808 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
18 | 1, 13, 17 | 3eqtri 2763 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
19 | 18 | breq2i 5047 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
20 | 3z 12175 | . . 3 ⊢ 3 ∈ ℤ | |
21 | 7 | nn0zi 12167 | . . . 4 ⊢ 𝐴 ∈ ℤ |
22 | 15 | nn0zi 12167 | . . . 4 ⊢ 𝐵 ∈ ℤ |
23 | zaddcl 12182 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
24 | 21, 22, 23 | mp2an 692 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
25 | 9nn 11893 | . . . . . 6 ⊢ 9 ∈ ℕ | |
26 | 25 | nnzi 12166 | . . . . 5 ⊢ 9 ∈ ℤ |
27 | zmulcl 12191 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
28 | 26, 21, 27 | mp2an 692 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
29 | zmulcl 12191 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
30 | 20, 21, 29 | mp2an 692 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
31 | dvdsmul1 15802 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
32 | 20, 30, 31 | mp2an 692 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
33 | 3t3e9 11962 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2745 | . . . . . . 7 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 7201 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
36 | 3cn 11876 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
37 | 36, 36, 8 | mulassi 10809 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
38 | 35, 37 | eqtri 2759 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
39 | 32, 38 | breqtrri 5066 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
40 | 28, 39 | pm3.2i 474 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
41 | dvdsadd2b 15830 | . . 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 1463 | . 2 ⊢ (3 ∥ (𝐴 + 𝐵) ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
43 | 19, 42 | bitr4i 281 | 1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 3c3 11851 9c9 11857 ℕ0cn0 12055 ℤcz 12141 ;cdc 12258 ∥ cdvds 15778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-dvds 15779 |
This theorem is referenced by: 257prm 44629 139prmALT 44664 31prm 44665 |
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