Theorem nnmcl 5975

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 5444	. . . . 5 ⊢ (x = B → (A ·𝑜 x) = (A ·𝑜 B))
2	1	eleq1d 2088	. . . 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 5444	. . . . 5 ⊢ (x = ∅ → (A ·𝑜 x) = (A ·𝑜 ∅))
5	4	eleq1d 2088	. . . 4 ⊢ (x = ∅ → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 ∅) ∈ 𝜔))
6		oveq2 5444	. . . . 5 ⊢ (x = y → (A ·𝑜 x) = (A ·𝑜 y))
7	6	eleq1d 2088	. . . 4 ⊢ (x = y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 y) ∈ 𝜔))
8		oveq2 5444	. . . . 5 ⊢ (x = suc y → (A ·𝑜 x) = (A ·𝑜 suc y))
9	8	eleq1d 2088	. . . 4 ⊢ (x = suc y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 suc y) ∈ 𝜔))
10		nnm0 5969	. . . . 5 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) = ∅)
11		peano1 4244	. . . . 5 ⊢ ∅ ∈ 𝜔
12	10, 11	syl6eqel 2110	. . . 4 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) ∈ 𝜔)
13		nnacl 5974	. . . . . . . 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 5971	. . . . . . 7 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → (A ·𝑜 suc y) = ((A ·𝑜 y) +𝑜 A))
17	16	eleq1d 2088	. . . . . 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 4251	. . 3 ⊢ (x ∈ 𝜔 → (A ∈ 𝜔 → (A ·𝑜 x) ∈ 𝜔))
21	3, 20	vtoclga 2596	. 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 1228   ∈ wcel 1374  ∅c0 3201  suc csuc 4051  𝜔com 4240  (class class class)co 5436   +_𝑜 coa 5913   ·_𝑜 comu 5914

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921

This theorem is referenced by:  nnmcli  5977  nndi  5980  nnmass  5981  nnmsucr  5982  nnmordi  6000  nnmord  6001  nnmword  6002  mulclpi  6188  enq0tr  6289  addcmpblnq0  6298  mulcmpblnq0  6299  mulcanenq0ec  6300  addclnq0  6306  mulclnq0  6307  nqpnq0nq  6308  distrnq0  6314  addassnq0lemcl  6316  addassnq0  6317