Theorem nnmcl 5999

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 5463	. . . . 5 ⊢ (x = B → (A ·𝑜 x) = (A ·𝑜 B))
2	1	eleq1d 2103	. . . 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 5463	. . . . 5 ⊢ (x = ∅ → (A ·𝑜 x) = (A ·𝑜 ∅))
5	4	eleq1d 2103	. . . 4 ⊢ (x = ∅ → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 ∅) ∈ 𝜔))
6		oveq2 5463	. . . . 5 ⊢ (x = y → (A ·𝑜 x) = (A ·𝑜 y))
7	6	eleq1d 2103	. . . 4 ⊢ (x = y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 y) ∈ 𝜔))
8		oveq2 5463	. . . . 5 ⊢ (x = suc y → (A ·𝑜 x) = (A ·𝑜 suc y))
9	8	eleq1d 2103	. . . 4 ⊢ (x = suc y → ((A ·𝑜 x) ∈ 𝜔 ↔ (A ·𝑜 suc y) ∈ 𝜔))
10		nnm0 5993	. . . . 5 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) = ∅)
11		peano1 4260	. . . . 5 ⊢ ∅ ∈ 𝜔
12	10, 11	syl6eqel 2125	. . . 4 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) ∈ 𝜔)
13		nnacl 5998	. . . . . . . 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 5995	. . . . . . 7 ⊢ ((A ∈ 𝜔 ∧ y ∈ 𝜔) → (A ·𝑜 suc y) = ((A ·𝑜 y) +𝑜 A))
17	16	eleq1d 2103	. . . . . 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 4267	. . 3 ⊢ (x ∈ 𝜔 → (A ∈ 𝜔 → (A ·𝑜 x) ∈ 𝜔))
21	3, 20	vtoclga 2613	. 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 1242   ∈ wcel 1390  ∅c0 3218  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +_𝑜 coa 5937   ·_𝑜 comu 5938

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945

This theorem is referenced by:  nnmcli  6001  nndi  6004  nnmass  6005  nnmsucr  6006  nnmordi  6025  nnmord  6026  nnmword  6027  mulclpi  6312  enq0tr  6416  addcmpblnq0  6425  mulcmpblnq0  6426  mulcanenq0ec  6427  addclnq0  6433  mulclnq0  6434  nqpnq0nq  6435  distrnq0  6441  addassnq0lemcl  6443  addassnq0  6444