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Theorem nnaddcl 8010
Description: Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
Assertion
Ref Expression
nnaddcl ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Proof of Theorem nnaddcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5548 . . . . 5 (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1))
21eleq1d 2122 . . . 4 (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ))
32imbi2d 223 . . 3 (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)))
4 oveq2 5548 . . . . 5 (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
54eleq1d 2122 . . . 4 (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ))
65imbi2d 223 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ)))
7 oveq2 5548 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1)))
87eleq1d 2122 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
98imbi2d 223 . . 3 (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
10 oveq2 5548 . . . . 5 (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
1110eleq1d 2122 . . . 4 (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ))
1211imbi2d 223 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)))
13 peano2nn 8002 . . 3 (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
14 peano2nn 8002 . . . . . 6 ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ)
15 nncn 7998 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
16 nncn 7998 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17 ax-1cn 7035 . . . . . . . . 9 1 ∈ ℂ
18 addass 7069 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
1917, 18mp3an3 1232 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
2015, 16, 19syl2an 277 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
2120eleq1d 2122 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
2214, 21syl5ib 147 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))
2322expcom 113 . . . 4 (𝑦 ∈ ℕ → (𝐴 ∈ ℕ → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
2423a2d 26 . . 3 (𝑦 ∈ ℕ → ((𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ) → (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
253, 6, 9, 12, 13, 24nnind 8006 . 2 (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))
2625impcom 120 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  (class class class)co 5540  cc 6945  1c1 6948   + caddc 6950  cn 7990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-addrcl 7039  ax-addass 7044
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-inn 7991
This theorem is referenced by:  nnmulcl  8011  nn2ge  8022  nnaddcld  8037  nnnn0addcl  8269  nn0addcl  8274  9p1e10  8429
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