Theorem nnmcl 6060

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 5520	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
2	1	eleq1d 2106	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝐵) ∈ ω))
3	2	imbi2d 219	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω)))
4		oveq2 5520	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
5	4	eleq1d 2106	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 ∅) ∈ ω))
6		oveq2 5520	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
7	6	eleq1d 2106	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝑦) ∈ ω))
8		oveq2 5520	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
9	8	eleq1d 2106	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 suc 𝑦) ∈ ω))
10		nnm0 6054	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
11		peano1 4317	. . . . 5 ⊢ ∅ ∈ ω
12	10, 11	syl6eqel 2128	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
13		nnacl 6059	. . . . . . . 8 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω)
14	13	expcom 109	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
15	14	adantr 261	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
16		nnmsuc 6056	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
17	16	eleq1d 2106	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
18	15, 17	sylibrd 158	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω))
19	18	expcom 109	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω)))
20	5, 7, 9, 12, 19	finds2 4324	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω))
21	3, 20	vtoclga 2619	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω))
22	21	impcom 116	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∅c0 3224  suc csuc 4102  ωcom 4313  (class class class)co 5512   +_𝑜 coa 5998   ·_𝑜 comu 5999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006

This theorem is referenced by:  nnmcli  6062  nndi  6065  nnmass  6066  nnmsucr  6067  nnmordi  6089  nnmord  6090  nnmword  6091  mulclpi  6426  enq0tr  6532  addcmpblnq0  6541  mulcmpblnq0  6542  mulcanenq0ec  6543  addclnq0  6549  mulclnq0  6550  nqpnq0nq  6551  distrnq0  6557  addassnq0lemcl  6559  addassnq0  6560