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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 12736 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
| 2 | 9p1e10 12735 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
| 3 | 2 | eqcomi 2746 | . . . . . . 7 ⊢ ;10 = (9 + 1) |
| 4 | 3 | oveq1i 7441 | . . . . . 6 ⊢ (;10 · 𝐴) = ((9 + 1) · 𝐴) |
| 5 | 9cn 12366 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
| 6 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 7 | 3dvdsdec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
| 8 | 7 | nn0cni 12538 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
| 9 | 5, 6, 8 | adddiri 11274 | . . . . . 6 ⊢ ((9 + 1) · 𝐴) = ((9 · 𝐴) + (1 · 𝐴)) |
| 10 | 8 | mullidi 11266 | . . . . . . 7 ⊢ (1 · 𝐴) = 𝐴 |
| 11 | 10 | oveq2i 7442 | . . . . . 6 ⊢ ((9 · 𝐴) + (1 · 𝐴)) = ((9 · 𝐴) + 𝐴) |
| 12 | 4, 9, 11 | 3eqtri 2769 | . . . . 5 ⊢ (;10 · 𝐴) = ((9 · 𝐴) + 𝐴) |
| 13 | 12 | oveq1i 7441 | . . . 4 ⊢ ((;10 · 𝐴) + 𝐵) = (((9 · 𝐴) + 𝐴) + 𝐵) |
| 14 | 5, 8 | mulcli 11268 | . . . . 5 ⊢ (9 · 𝐴) ∈ ℂ |
| 15 | 3dvdsdec.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
| 16 | 15 | nn0cni 12538 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
| 17 | 14, 8, 16 | addassi 11271 | . . . 4 ⊢ (((9 · 𝐴) + 𝐴) + 𝐵) = ((9 · 𝐴) + (𝐴 + 𝐵)) |
| 18 | 1, 13, 17 | 3eqtri 2769 | . . 3 ⊢ ;𝐴𝐵 = ((9 · 𝐴) + (𝐴 + 𝐵)) |
| 19 | 18 | breq2i 5151 | . 2 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ ((9 · 𝐴) + (𝐴 + 𝐵))) |
| 20 | 3z 12650 | . . 3 ⊢ 3 ∈ ℤ | |
| 21 | 7 | nn0zi 12642 | . . . 4 ⊢ 𝐴 ∈ ℤ |
| 22 | 15 | nn0zi 12642 | . . . 4 ⊢ 𝐵 ∈ ℤ |
| 23 | zaddcl 12657 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
| 24 | 21, 22, 23 | mp2an 692 | . . 3 ⊢ (𝐴 + 𝐵) ∈ ℤ |
| 25 | 9nn 12364 | . . . . . 6 ⊢ 9 ∈ ℕ | |
| 26 | 25 | nnzi 12641 | . . . . 5 ⊢ 9 ∈ ℤ |
| 27 | zmulcl 12666 | . . . . 5 ⊢ ((9 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (9 · 𝐴) ∈ ℤ) | |
| 28 | 26, 21, 27 | mp2an 692 | . . . 4 ⊢ (9 · 𝐴) ∈ ℤ |
| 29 | zmulcl 12666 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (3 · 𝐴) ∈ ℤ) | |
| 30 | 20, 21, 29 | mp2an 692 | . . . . . 6 ⊢ (3 · 𝐴) ∈ ℤ |
| 31 | dvdsmul1 16315 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ (3 · 𝐴) ∈ ℤ) → 3 ∥ (3 · (3 · 𝐴))) | |
| 32 | 20, 30, 31 | mp2an 692 | . . . . 5 ⊢ 3 ∥ (3 · (3 · 𝐴)) |
| 33 | 3t3e9 12433 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
| 34 | 33 | eqcomi 2746 | . . . . . . 7 ⊢ 9 = (3 · 3) |
| 35 | 34 | oveq1i 7441 | . . . . . 6 ⊢ (9 · 𝐴) = ((3 · 3) · 𝐴) |
| 36 | 3cn 12347 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 37 | 36, 36, 8 | mulassi 11272 | . . . . . 6 ⊢ ((3 · 3) · 𝐴) = (3 · (3 · 𝐴)) |
| 38 | 35, 37 | eqtri 2765 | . . . . 5 ⊢ (9 · 𝐴) = (3 · (3 · 𝐴)) |
| 39 | 32, 38 | breqtrri 5170 | . . . 4 ⊢ 3 ∥ (9 · 𝐴) |
| 40 | 28, 39 | pm3.2i 470 | . . 3 ⊢ ((9 · 𝐴) ∈ ℤ ∧ 3 ∥ (9 · 𝐴)) |
| 41 | dvdsadd2b 16343 | . . 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 278 | 1 ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 3c3 12322 9c9 12328 ℕ0cn0 12526 ℤcz 12613 ;cdc 12733 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-dvds 16291 |
| This theorem is referenced by: 257prm 47548 139prmALT 47583 31prm 47584 |
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