Theorem nnmcl 6534

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 5926	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
2	1	eleq1d 2262	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝐵) ∈ ω))
3	2	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω)))
4		oveq2 5926	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
5	4	eleq1d 2262	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o ∅) ∈ ω))
6		oveq2 5926	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
7	6	eleq1d 2262	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝑦) ∈ ω))
8		oveq2 5926	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
9	8	eleq1d 2262	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o suc 𝑦) ∈ ω))
10		nnm0 6528	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
11		peano1 4626	. . . . 5 ⊢ ∅ ∈ ω
12	10, 11	eqeltrdi 2284	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
13		nnacl 6533	. . . . . . . 8 ⊢ (((𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω)
14	13	expcom 116	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
15	14	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
16		nnmsuc 6530	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
17	16	eleq1d 2262	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) ∈ ω ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
18	15, 17	sylibrd 169	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω))
19	18	expcom 116	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω)))
20	5, 7, 9, 12, 19	finds2 4633	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω))
21	3, 20	vtoclga 2826	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω))
22	21	impcom 125	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∅c0 3446  suc csuc 4396  ωcom 4622  (class class class)co 5918   +_o coa 6466   ·_o comu 6467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474

This theorem is referenced by:  nnmcli  6536  nndi  6539  nnmass  6540  nnmsucr  6541  nnmordi  6569  nnmord  6570  nnmword  6571  mulclpi  7388  enq0tr  7494  addcmpblnq0  7503  mulcmpblnq0  7504  mulcanenq0ec  7505  addclnq0  7511  mulclnq0  7512  nqpnq0nq  7513  distrnq0  7519  addassnq0lemcl  7521  addassnq0  7522