MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnacl Structured version   Visualization version   GIF version

Theorem nnacl 7688
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 6655 . . . . 5 (𝑥 = 𝐵 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐵))
21eleq1d 2685 . . . 4 (𝑥 = 𝐵 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝐵) ∈ ω))
32imbi2d 330 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω)))
4 oveq2 6655 . . . . 5 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
54eleq1d 2685 . . . 4 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 ∅) ∈ ω))
6 oveq2 6655 . . . . 5 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
76eleq1d 2685 . . . 4 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 𝑦) ∈ ω))
8 oveq2 6655 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
98eleq1d 2685 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ ω ↔ (𝐴 +𝑜 suc 𝑦) ∈ ω))
10 nna0 7681 . . . . . 6 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1110eleq1d 2685 . . . . 5 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ ω ↔ 𝐴 ∈ ω))
1211ibir 257 . . . 4 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) ∈ ω)
13 peano2 7083 . . . . . 6 ((𝐴 +𝑜 𝑦) ∈ ω → suc (𝐴 +𝑜 𝑦) ∈ ω)
14 nnasuc 7683 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
1514eleq1d 2685 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 suc 𝑦) ∈ ω ↔ suc (𝐴 +𝑜 𝑦) ∈ ω))
1613, 15syl5ibr 236 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω))
1716expcom 451 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 suc 𝑦) ∈ ω)))
185, 7, 9, 12, 17finds2 7091 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝑥) ∈ ω))
193, 18vtoclga 3270 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 +𝑜 𝐵) ∈ ω))
2019impcom 446 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  c0 3913  suc csuc 5723  (class class class)co 6647  ωcom 7062   +𝑜 coa 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-oadd 7561
This theorem is referenced by:  nnmcl  7689  nnacli  7691  nnarcl  7693  nnaord  7696  nnawordi  7698  nnaass  7699  nndi  7700  nnaword  7704  nnawordex  7714  oaabslem  7720  unfilem1  8221  unfi  8224  nnacda  9020  ficardun  9021  ficardun2  9022  pwsdompw  9023  addclpi  9711  hashgadd  13161  hashdom  13163  finxpreclem4  33211
  Copyright terms: Public domain W3C validator