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Mirrors > Home > ILE Home > Th. List > nnnn0addcl | GIF version |
Description: A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nnnn0addcl | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9003 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nnaddcl 8764 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
3 | oveq2 5790 | . . . . 5 ⊢ (𝑁 = 0 → (𝑀 + 𝑁) = (𝑀 + 0)) | |
4 | nncn 8752 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ) | |
5 | 4 | addid1d 7935 | . . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀 + 0) = 𝑀) |
6 | 3, 5 | sylan9eqr 2195 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 = 0) → (𝑀 + 𝑁) = 𝑀) |
7 | simpl 108 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 = 0) → 𝑀 ∈ ℕ) | |
8 | 6, 7 | eqeltrd 2217 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 = 0) → (𝑀 + 𝑁) ∈ ℕ) |
9 | 2, 8 | jaodan 787 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑀 + 𝑁) ∈ ℕ) |
10 | 1, 9 | sylan2b 285 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 0cc0 7644 + caddc 7647 ℕcn 8744 ℕ0cn0 9001 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 df-n0 9002 |
This theorem is referenced by: nn0nnaddcl 9032 elz2 9146 bcxmas 11290 |
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