ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnaddcld GIF version

Theorem nnaddcld 8142
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 8115 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
41, 2, 3syl2anc 403 1 (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  (class class class)co 5537   + caddc 7035  cn 8095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-addrcl 7124  ax-addass 7129
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540  df-inn 8096
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator