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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 9044 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
| 2 | nn0cn 9305 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
| 3 | addcom 8209 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) | |
| 4 | 1, 2, 3 | syl2an 289 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) |
| 5 | nnnn0addcl 9325 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) ∈ ℕ) | |
| 6 | 4, 5 | eqeltrrd 2283 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) |
| 7 | 6 | ancoms 268 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2176 (class class class)co 5944 ℂcc 7923 + caddc 7928 ℕcn 9036 ℕ0cn0 9295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 ax-sep 4162 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-i2m1 8030 ax-0id 8033 ax-rnegex 8034 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-iota 5232 df-fv 5279 df-ov 5947 df-inn 9037 df-n0 9296 |
| This theorem is referenced by: nn0p1nn 9334 nnaddm1cl 9434 numnncl 9513 modfzo0difsn 10540 |
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