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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 979 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) | |
2 | 3simpb 980 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
3 | zaddcl 9213 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) | |
4 | 3 | anim2i 340 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐾 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ)) |
5 | 4 | 3impb 1181 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ)) |
6 | zaddcl 9213 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
7 | 6 | adantl 275 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 + 𝑦) ∈ ℤ) |
8 | zcn 9178 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
9 | zcn 9178 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
10 | zcn 9178 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
11 | adddir 7872 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) | |
12 | 8, 9, 10, 11 | syl3an 1262 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
13 | 12 | 3comr 1193 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
14 | 13 | 3expb 1186 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 + 𝑦) · 𝐾) = ((𝑥 · 𝐾) + (𝑦 · 𝐾))) |
15 | oveq12 5836 | . . . . 5 ⊢ (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝐾) = 𝑁) → ((𝑥 · 𝐾) + (𝑦 · 𝐾)) = (𝑀 + 𝑁)) | |
16 | 14, 15 | sylan9eq 2210 | . . . 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 1144 | . 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 11711 | 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 963 = wceq 1335 ∈ wcel 2128 class class class wbr 3967 (class class class)co 5827 ℂcc 7733 + caddc 7738 · cmul 7740 ℤcz 9173 ∥ cdvds 11695 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-dvds 11696 |
This theorem is referenced by: dvdssub2 11742 dvdsadd2b 11747 bezoutlemstep 11897 bezoutlembi 11905 dvdsmulgcd 11925 bezoutr 11932 pythagtriplem19 12173 |
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