Theorem nnmcl 6649

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 6026	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
2	1	eleq1d 2300	. . . 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 6026	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
5	4	eleq1d 2300	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o ∅) ∈ ω))
6		oveq2 6026	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
7	6	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝑦) ∈ ω))
8		oveq2 6026	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
9	8	eleq1d 2300	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o suc 𝑦) ∈ ω))
10		nnm0 6643	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
11		peano1 4692	. . . . 5 ⊢ ∅ ∈ ω
12	10, 11	eqeltrdi 2322	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
13		nnacl 6648	. . . . . . . 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 6645	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
17	16	eleq1d 2300	. . . . . 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 4699	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω))
21	3, 20	vtoclga 2870	. 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 1397   ∈ wcel 2202  ∅c0 3494  suc csuc 4462  ωcom 4688  (class class class)co 6018   +_o coa 6579   ·_o comu 6580

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-oadd 6586  df-omul 6587

This theorem is referenced by:  nnmcli  6651  nndi  6654  nnmass  6655  nnmsucr  6656  nnmordi  6684  nnmord  6685  nnmword  6686  mulclpi  7548  enq0tr  7654  addcmpblnq0  7663  mulcmpblnq0  7664  mulcanenq0ec  7665  addclnq0  7671  mulclnq0  7672  nqpnq0nq  7673  distrnq0  7679  addassnq0lemcl  7681  addassnq0  7682