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Theorem nnacl 7578
Description: Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
nnacl ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)

Proof of Theorem nnacl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . 5 (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
21eleq1d 2672 . . . 4 (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝐵) ∈ ω))
32imbi2d 329 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω)))
4 oveq2 6557 . . . . 5 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
54eleq1d 2672 . . . 4 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 ∅) ∈ ω))
6 oveq2 6557 . . . . 5 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
76eleq1d 2672 . . . 4 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝑦) ∈ ω))
8 oveq2 6557 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
98eleq1d 2672 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 suc 𝑦) ∈ ω))
10 nna0 7571 . . . . . 6 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1110eleq1d 2672 . . . . 5 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ ω ↔ 𝐴 ∈ ω))
1211ibir 256 . . . 4 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) ∈ ω)
13 peano2 6978 . . . . . 6 ((𝐴 +𝑜 𝑦) ∈ ω → suc (𝐴 +𝑜 𝑦) ∈ ω)
14 nnasuc 7573 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
1514eleq1d 2672 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 suc 𝑦) ∈ ω ↔ suc (𝐴 +𝑜 𝑦) ∈ ω))
1613, 15syl5ibr 235 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω))
1716expcom 450 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω)))
185, 7, 9, 12, 17finds2 6986 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω))
193, 18vtoclga 3245 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω))
2019impcom 445 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  c0 3874  suc csuc 5642  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451
This theorem is referenced by:  nnmcl  7579  nnacli  7581  nnarcl  7583  nnaord  7586  nnawordi  7588  nnaass  7589  nndi  7590  nnaword  7594  nnawordex  7604  oaabslem  7610  unfilem1  8109  unfi  8112  nnacda  8906  ficardun  8907  ficardun2  8908  pwsdompw  8909  addclpi  9593  hashgadd  13027  hashdom  13029  finxpreclem4  32407
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