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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 8856 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | nn0cn 9115 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
3 | addcom 8026 | . . . 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 9135 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) ∈ ℕ) | |
6 | 4, 5 | eqeltrrd 2242 | . 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 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 + caddc 7747 ℕcn 8848 ℕ0cn0 9105 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-inn 8849 df-n0 9106 |
This theorem is referenced by: nn0p1nn 9144 nnaddm1cl 9243 numnncl 9322 modfzo0difsn 10320 |
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