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Theorem nnm0r 7930
Description: Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnm0r (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)

Proof of Theorem nnm0r
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6886 . . 3 (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
21eqeq1d 2801 . 2 (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 6886 . . 3 (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
43eqeq1d 2801 . 2 (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5 oveq2 6886 . . 3 (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
65eqeq1d 2801 . 2 (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7 oveq2 6886 . . 3 (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
87eqeq1d 2801 . 2 (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9 0elon 5994 . . 3 ∅ ∈ On
10 om0 7837 . . 3 (∅ ∈ On → (∅ ·𝑜 ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·𝑜 ∅) = ∅
12 oveq1 6885 . . . 4 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
13 oa0 7836 . . . . 5 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
149, 13ax-mp 5 . . . 4 (∅ +𝑜 ∅) = ∅
1512, 14syl6eq 2849 . . 3 ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅)
16 peano1 7319 . . . . 5 ∅ ∈ ω
17 nnmsuc 7927 . . . . 5 ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1816, 17mpan 682 . . . 4 (𝑦 ∈ ω → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
1918eqeq1d 2801 . . 3 (𝑦 ∈ ω → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅))
2015, 19syl5ibr 238 . 2 (𝑦 ∈ ω → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
212, 4, 6, 8, 11, 20finds 7326 1 (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  c0 4115  Oncon0 5941  suc csuc 5943  (class class class)co 6878  ωcom 7299   +𝑜 coa 7796   ·𝑜 comu 7797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  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 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-oadd 7803  df-omul 7804
This theorem is referenced by:  nnmcom  7946  nnmord  7952  nnmwordi  7955
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