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Theorem nnmcl 7929
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 6882 . . . . 5 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
21eleq1d 2870 . . . 4 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝐵) ∈ ω))
32imbi2d 331 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω)))
4 oveq2 6882 . . . . 5 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
54eleq1d 2870 . . . 4 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 ∅) ∈ ω))
6 oveq2 6882 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
76eleq1d 2870 . . . 4 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 𝑦) ∈ ω))
8 oveq2 6882 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
98eleq1d 2870 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ ω ↔ (𝐴 ·𝑜 suc 𝑦) ∈ ω))
10 nnm0 7922 . . . . 5 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
11 peano1 7315 . . . . 5 ∅ ∈ ω
1210, 11syl6eqel 2893 . . . 4 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
13 nnacl 7928 . . . . . . . 8 (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω)
1413expcom 400 . . . . . . 7 (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
1514adantr 468 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
16 nnmsuc 7924 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
1716eleq1d 2870 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ ω))
1815, 17sylibrd 250 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω))
1918expcom 400 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ·𝑜 𝑦) ∈ ω → (𝐴 ·𝑜 suc 𝑦) ∈ ω)))
205, 7, 9, 12, 19finds2 7324 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝑥) ∈ ω))
213, 20vtoclga 3465 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ·𝑜 𝐵) ∈ ω))
2221impcom 396 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  c0 4116  suc csuc 5938  (class class class)co 6874  ωcom 7295   +𝑜 coa 7793   ·𝑜 comu 7794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-oadd 7800  df-omul 7801
This theorem is referenced by:  nnecl  7930  nnmcli  7932  nndi  7940  nnmass  7941  nnmsucr  7942  nnmordi  7948  nnmord  7949  nnmword  7950  omabslem  7963  nnneo  7968  nneob  7969  fin1a2lem4  9510  mulclpi  10000
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