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Theorem nnmcl 8649
Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. Theorem 2.20 of [Schloeder] p. 6. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
nnmcl ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)

Proof of Theorem nnmcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . 5 (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
21eleq1d 2824 . . . 4 (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝐵) ∈ ω))
32imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω)))
4 oveq2 7439 . . . . 5 (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
54eleq1d 2824 . . . 4 (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o ∅) ∈ ω))
6 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
76eleq1d 2824 . . . 4 (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝑦) ∈ ω))
8 oveq2 7439 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
98eleq1d 2824 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o suc 𝑦) ∈ ω))
10 nnm0 8642 . . . . 5 (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
11 peano1 7911 . . . . 5 ∅ ∈ ω
1210, 11eqeltrdi 2847 . . . 4 (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
13 nnacl 8648 . . . . . . . 8 (((𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω)
1413expcom 413 . . . . . . 7 (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
1514adantr 480 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
16 nnmsuc 8644 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
1716eleq1d 2824 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) ∈ ω ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
1815, 17sylibrd 259 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω))
1918expcom 413 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω)))
205, 7, 9, 12, 19finds2 7921 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω))
213, 20vtoclga 3577 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω))
2221impcom 407 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  c0 4339  suc csuc 6388  (class class class)co 7431  ωcom 7887   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510
This theorem is referenced by:  nnecl  8650  nnmcli  8652  nndi  8660  nnmass  8661  nnmsucr  8662  nnmordi  8668  nnmord  8669  nnmword  8670  omabslem  8687  nnneo  8692  nneob  8693  fin1a2lem4  10441  mulclpi  10931  nnamecl  43277
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