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Mirrors > Home > ILE Home > Th. List > nn0addcl | GIF version |
Description: Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
nn0addcl | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 8488 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | id 19 | . . 3 ⊢ (ℕ ⊆ ℂ → ℕ ⊆ ℂ) | |
3 | df-n0 8735 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
4 | nnaddcl 8503 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
5 | 4 | adantl 272 | . . 3 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑀 + 𝑁) ∈ ℕ) |
6 | 2, 3, 5 | un0addcl 8767 | . 2 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 + 𝑁) ∈ ℕ0) |
7 | 1, 6 | mpan 416 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1439 ⊆ wss 3000 (class class class)co 5666 ℂcc 7409 + caddc 7414 ℕcn 8483 ℕ0cn0 8734 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-i2m1 7511 ax-0id 7514 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 df-inn 8484 df-n0 8735 |
This theorem is referenced by: nn0addcli 8771 peano2nn0 8774 nn0addcld 8791 nn0readdcl 8793 difelfznle 9607 elfzodifsumelfzo 9673 expadd 10058 faclbnd6 10213 facavg 10215 fsumnn0cl 10858 bcxmas 10944 eftlub 11041 |
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