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Theorem nnaddcl 8938
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 5882 . . . . 5 (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1))
21eleq1d 2246 . . . 4 (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ))
32imbi2d 230 . . 3 (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)))
4 oveq2 5882 . . . . 5 (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
54eleq1d 2246 . . . 4 (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ))
65imbi2d 230 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ)))
7 oveq2 5882 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1)))
87eleq1d 2246 . . . 4 (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
98imbi2d 230 . . 3 (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
10 oveq2 5882 . . . . 5 (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
1110eleq1d 2246 . . . 4 (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ))
1211imbi2d 230 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)))
13 peano2nn 8930 . . 3 (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
14 peano2nn 8930 . . . . . 6 ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ)
15 nncn 8926 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
16 nncn 8926 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17 ax-1cn 7903 . . . . . . . . 9 1 ∈ ℂ
18 addass 7940 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
1917, 18mp3an3 1326 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
2015, 16, 19syl2an 289 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
2120eleq1d 2246 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
2214, 21imbitrid 154 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))
2322expcom 116 . . . 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 8934 . 2 (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))
2625impcom 125 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  (class class class)co 5874  cc 7808  1c1 7811   + caddc 7813  cn 8918
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4121  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-addrcl 7907  ax-addass 7912
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877  df-inn 8919
This theorem is referenced by:  nnmulcl  8939  nn2ge  8951  nnaddcld  8966  nnnn0addcl  9205  nn0addcl  9210  9p1e10  9385  pythagtriplem4  12267  mulgnndir  13010
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