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Mirrors > Home > ILE Home > Th. List > dvds2add | GIF version |
Description: If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds2add | ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ (𝑀 + 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 941 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) | |
2 | 3simpb 942 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
3 | zaddcl 8851 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) | |
4 | 3 | anim2i 335 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐾 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ)) |
5 | 4 | 3impb 1140 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ)) |
6 | zaddcl 8851 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
7 | 6 | adantl 272 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 + 𝑦) ∈ ℤ) |
8 | zcn 8816 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
9 | zcn 8816 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
10 | zcn 8816 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
11 | adddir 7540 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) | |
12 | 8, 9, 10, 11 | syl3an 1217 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
13 | 12 | 3comr 1152 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
14 | 13 | 3expb 1145 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
15 | oveq12 5675 | . . . . 5 ⊢ (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑥 · 𝐾) + (𝑦 · 𝐾)) = (𝑀 + 𝑁)) | |
16 | 14, 15 | sylan9eq 2141 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝐾) = 𝑁)) → ((𝑥 + 𝑦) · 𝐾) = (𝑀 + 𝑁)) |
17 | 16 | ex 114 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑥 + 𝑦) · 𝐾) = (𝑀 + 𝑁))) |
18 | 17 | 3ad2antl1 1106 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑥 + 𝑦) · 𝐾) = (𝑀 + 𝑁))) |
19 | 1, 2, 5, 7, 18 | dvds2lem 11147 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ (𝑀 + 𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℂcc 7409 + caddc 7414 · cmul 7416 ℤcz 8811 ∥ cdvds 11135 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812 df-dvds 11136 |
This theorem is referenced by: dvdssub2 11177 dvdsadd2b 11182 bezoutlemstep 11325 bezoutlembi 11333 dvdsmulgcd 11353 bezoutr 11360 |
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