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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 8949 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | id 19 | . . 3 ⊢ (ℕ ⊆ ℂ → ℕ ⊆ ℂ) | |
3 | df-n0 9202 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
4 | nnaddcl 8964 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
5 | 4 | adantl 277 | . . 3 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑀 + 𝑁) ∈ ℕ) |
6 | 2, 3, 5 | un0addcl 9234 | . 2 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 + 𝑁) ∈ ℕ0) |
7 | 1, 6 | mpan 424 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ⊆ wss 3144 (class class class)co 5892 ℂcc 7834 + caddc 7839 ℕcn 8944 ℕ0cn0 9201 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-i2m1 7941 ax-0id 7944 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5193 df-fv 5240 df-ov 5895 df-inn 8945 df-n0 9202 |
This theorem is referenced by: nn0addcli 9238 peano2nn0 9241 nn0addcld 9258 nn0readdcl 9260 difelfznle 10160 elfzodifsumelfzo 10226 expadd 10588 faclbnd6 10751 facavg 10753 fsumnn0cl 11438 bcxmas 11524 eftlub 11725 4sqlem1 12415 nn0subm 13879 2sqlem7 14906 |
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