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Mirrors > Home > ILE Home > Th. List > nn0nnaddcl | GIF version |
Description: A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Ref | Expression |
---|---|
nn0nnaddcl | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 8696 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | nn0cn 8955 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
3 | addcom 7867 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) | |
4 | 1, 2, 3 | syl2an 287 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) |
5 | nnnn0addcl 8975 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) ∈ ℕ) | |
6 | 4, 5 | eqeltrrd 2195 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) |
7 | 6 | ancoms 266 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 + caddc 7591 ℕcn 8688 ℕ0cn0 8945 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-inn 8689 df-n0 8946 |
This theorem is referenced by: nn0p1nn 8984 nnaddm1cl 9083 numnncl 9159 modfzo0difsn 10136 |
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