Theorem nnmcl 8582

Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. Theorem 2.20 of [Schloeder] p. 6. (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 7404	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
2	1	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝐵) ∈ ω))
3	2	imbi2d 342	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω)))
4		oveq2 7404	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
5	4	eleq1d 2847	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o ∅) ∈ ω))
6		oveq2 7404	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
7	6	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o 𝑦) ∈ ω))
8		oveq2 7404	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
9	8	eleq1d 2847	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ ω ↔ (𝐴 ·o suc 𝑦) ∈ ω))
10		nnm0 8575	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
11		peano1 7869	. . . . 5 ⊢ ∅ ∈ ω
12	10, 11	eqeltrdi 2870	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
13		nnacl 8581	. . . . . . . 8 ⊢ (((𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω)
14	13	expcom 417	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
15	14	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
16		nnmsuc 8577	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
17	16	eleq1d 2847	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) ∈ ω ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ ω))
18	15, 17	sylibrd 261	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω))
19	18	expcom 417	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·o 𝑦) ∈ ω → (𝐴 ·o suc 𝑦) ∈ ω)))
20	5, 7, 9, 12, 19	finds2 7879	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝑥) ∈ ω))
21	3, 20	vtoclga 3541	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·o 𝐵) ∈ ω))
22	21	impcom 411	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∅c0 4285  suc csuc 6348  (class class class)co 7396  ωcom 7846   +_o coa 8434   ·_o comu 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-oadd 8441  df-omul 8442

This theorem is referenced by:  nnecl  8583  nnmcli  8585  nndi  8593  nnmass  8594  nnmsucr  8595  nnmordi  8601  nnmord  8602  nnmword  8603  omabslem  8620  nnneo  8625  nneob  8626  fin1a2lem4  10360  mulclpi  10851  nnamecl  43864