MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnmcl Structured version   Visualization version   GIF version

Theorem nnmcl 7637
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.
StepHypRef Expression
1 oveq2 6612 . . . . 5 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
21eleq1d 2683 . . . 4 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝐵) ∈ ω))
32imbi2d 330 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω)))
4 oveq2 6612 . . . . 5 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
54eleq1d 2683 . . . 4 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 ∅) ∈ ω))
6 oveq2 6612 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
76eleq1d 2683 . . . 4 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝑦) ∈ ω))
8 oveq2 6612 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
98eleq1d 2683 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 suc 𝑦) ∈ ω))
10 nnm0 7630 . . . . 5 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
11 peano1 7032 . . . . 5 ∅ ∈ ω
1210, 11syl6eqel 2706 . . . 4 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
13 nnacl 7636 . . . . . . . 8 (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω)
1413expcom 451 . . . . . . 7 (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
1514adantr 481 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
16 nnmsuc 7632 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
1716eleq1d 2683 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
1815, 17sylibrd 249 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω))
1918expcom 451 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω)))
205, 7, 9, 12, 19finds2 7041 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω))
213, 20vtoclga 3258 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω))
2221impcom 446 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3891  suc csuc 5684  (class class class)co 6604  ωcom 7012   +𝑜 coa 7502   ·𝑜 comu 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510
This theorem is referenced by:  nnecl  7638  nnmcli  7640  nndi  7648  nnmass  7649  nnmsucr  7650  nnmordi  7656  nnmord  7657  nnmword  7658  omabslem  7671  nnneo  7676  nneob  7677  fin1a2lem4  9169  mulclpi  9659
  Copyright terms: Public domain W3C validator