Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > zdivadd | GIF version |
Description: Property of divisibility: if 𝐷 divides 𝐴 and 𝐵 then it divides 𝐴 + 𝐵. (Contributed by NM, 3-Oct-2008.) |
Ref | Expression |
---|---|
zdivadd | ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9204 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | zcn 9204 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
3 | nncn 8873 | . . . . . 6 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℂ) | |
4 | nnap0 8894 | . . . . . 6 ⊢ (𝐷 ∈ ℕ → 𝐷 # 0) | |
5 | 3, 4 | jca 304 | . . . . 5 ⊢ (𝐷 ∈ ℕ → (𝐷 ∈ ℂ ∧ 𝐷 # 0)) |
6 | divdirap 8601 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 + 𝐵) / 𝐷) = ((𝐴 / 𝐷) + (𝐵 / 𝐷))) | |
7 | 1, 2, 5, 6 | syl3an 1275 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ((𝐴 + 𝐵) / 𝐷) = ((𝐴 / 𝐷) + (𝐵 / 𝐷))) |
8 | 7 | 3comr 1206 | . . 3 ⊢ ((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 𝐵) / 𝐷) = ((𝐴 / 𝐷) + (𝐵 / 𝐷))) |
9 | 8 | adantr 274 | . 2 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) = ((𝐴 / 𝐷) + (𝐵 / 𝐷))) |
10 | zaddcl 9239 | . . 3 ⊢ (((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ) → ((𝐴 / 𝐷) + (𝐵 / 𝐷)) ∈ ℤ) | |
11 | 10 | adantl 275 | . 2 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 / 𝐷) + (𝐵 / 𝐷)) ∈ ℤ) |
12 | 9, 11 | eqeltrd 2247 | 1 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 ℂcc 7759 0cc0 7761 + caddc 7764 # cap 8487 / cdiv 8576 ℕcn 8865 ℤcz 9199 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-n0 9123 df-z 9200 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |