Theorem nnmcl 7689

Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (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.

Step	Hyp	Ref	Expression
1		oveq2 6655	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
2	1	eleq1d 2685	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝐵) ∈ ω))
3	2	imbi2d 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 ∅))
5	4	eleq1d 2685	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 ∅) ∈ ω))
6		oveq2 6655	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
7	6	eleq1d 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))
9	8	eleq1d 2685	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 suc 𝑦) ∈ ω))
10		nnm0 7682	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
11		peano1 7082	. . . . 5 ⊢ ∅ ∈ ω
12	10, 11	syl6eqel 2708	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
13		nnacl 7688	. . . . . . . 8 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω)
14	13	expcom 451	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
15	14	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
16		nnmsuc 7684	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
17	16	eleq1d 2685	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
18	15, 17	sylibrd 249	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω))
19	18	expcom 451	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω)))
20	5, 7, 9, 12, 19	finds2 7091	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω))
21	3, 20	vtoclga 3270	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω))
22	21	impcom 446	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)

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   ·_𝑜 comu 7555

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  df-omul 7562

This theorem is referenced by:  nnecl  7690  nnmcli  7692  nndi  7700  nnmass  7701  nnmsucr  7702  nnmordi  7708  nnmord  7709  nnmword  7710  omabslem  7723  nnneo  7728  nneob  7729  fin1a2lem4  9222  mulclpi  9712