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Theorem nnacl 8229
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 7156 . . . . 5 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
21eleq1d 2895 . . . 4 (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o 𝐵) ∈ ω))
32imbi2d 343 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +o 𝐵) ∈ ω)))
4 oveq2 7156 . . . . 5 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
54eleq1d 2895 . . . 4 (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o ∅) ∈ ω))
6 oveq2 7156 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
76eleq1d 2895 . . . 4 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o 𝑦) ∈ ω))
8 oveq2 7156 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
98eleq1d 2895 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ ω ↔ (𝐴 +o suc 𝑦) ∈ ω))
10 nna0 8222 . . . . . 6 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
1110eleq1d 2895 . . . . 5 (𝐴 ∈ ω → ((𝐴 +o ∅) ∈ ω ↔ 𝐴 ∈ ω))
1211ibir 270 . . . 4 (𝐴 ∈ ω → (𝐴 +o ∅) ∈ ω)
13 peano2 7594 . . . . . 6 ((𝐴 +o 𝑦) ∈ ω → suc (𝐴 +o 𝑦) ∈ ω)
14 nnasuc 8224 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
1514eleq1d 2895 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o suc 𝑦) ∈ ω ↔ suc (𝐴 +o 𝑦) ∈ ω))
1613, 15syl5ibr 248 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o suc 𝑦) ∈ ω))
1716expcom 416 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o suc 𝑦) ∈ ω)))
185, 7, 9, 12, 17finds2 7602 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +o 𝑥) ∈ ω))
193, 18vtoclga 3572 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +o 𝐵) ∈ ω))
2019impcom 410 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  c0 4289  suc csuc 6186  (class class class)co 7148  ωcom 7572   +o coa 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098
This theorem is referenced by:  nnmcl  8230  nnacli  8232  nnarcl  8234  nnaord  8237  nnawordi  8239  nnaass  8240  nndi  8241  nnaword  8245  nnawordex  8255  oaabslem  8262  unfilem1  8774  unfi  8777  nnadju  9615  ficardun  9616  ficardun2  9617  pwsdompw  9618  addclpi  10306  hashgadd  13730  hashdom  13732  finxpreclem4  34662
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