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Theorem nnmcl 8650
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 2826 . . . 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 2826 . . . 4 (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o ∅) ∈ ω))
6 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
76eleq1d 2826 . . . 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 2826 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o suc 𝑦) ∈ ω))
10 nnm0 8643 . . . . 5 (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
11 peano1 7910 . . . . 5 ∅ ∈ ω
1210, 11eqeltrdi 2849 . . . 4 (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
13 nnacl 8649 . . . . . . . 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 8645 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
1716eleq1d 2826 . . . . . 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 7920 . . 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 1540  wcel 2108  c0 4333  suc csuc 6386  (class class class)co 7431  ωcom 7887   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511
This theorem is referenced by:  nnecl  8651  nnmcli  8653  nndi  8661  nnmass  8662  nnmsucr  8663  nnmordi  8669  nnmord  8670  nnmword  8671  omabslem  8688  nnneo  8693  nneob  8694  fin1a2lem4  10443  mulclpi  10933  nnamecl  43300
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