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Theorem nnaddcld 7961
Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
nnge1d.1 (𝜑𝐴 ∈ ℕ)
nnmulcld.2 (𝜑𝐵 ∈ ℕ)
Assertion
Ref Expression
nnaddcld (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Proof of Theorem nnaddcld
StepHypRef Expression
1 nnge1d.1 . 2 (𝜑𝐴 ∈ ℕ)
2 nnmulcld.2 . 2 (𝜑𝐵 ∈ ℕ)
3 nnaddcl 7934 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
41, 2, 3syl2anc 391 1 (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  (class class class)co 5512   + caddc 6892  cn 7914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-addrcl 6981  ax-addass 6986
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7915
This theorem is referenced by: (None)
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