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Mirrors > Home > ILE Home > Th. List > nn0addcld | GIF version |
Description: Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
nn0addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0addcld | ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | nn0addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
3 | nn0addcl 9163 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℕ0) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 (class class class)co 5851 + caddc 7770 ℕ0cn0 9128 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-i2m1 7872 ax-0id 7875 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5854 df-inn 8872 df-n0 9129 |
This theorem is referenced by: modsumfzodifsn 10345 expaddzap 10513 nn0opthlem1d 10647 nn0opthlem2d 10648 nn0opthd 10649 nn0opth2d 10650 bccl 10694 mertenslemi1 11491 pcpremul 12240 gzabssqcl 12326 4sqlem2 12334 mul4sq 12339 2sqlem8 13718 |
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