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Theorem nnm0r 8228
 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 7154 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2826 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7154 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2826 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7154 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2826 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7154 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2826 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6232 . . 3 ∅ ∈ On
10 om0 8134 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7153 . . . 4 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 oa0 8133 . . . . 5 (∅ ∈ On → (∅ +o ∅) = ∅)
149, 13ax-mp 5 . . . 4 (∅ +o ∅) = ∅
1512, 14syl6eq 2875 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = ∅)
16 peano1 7592 . . . . 5 ∅ ∈ ω
17 nnmsuc 8225 . . . . 5 ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
1816, 17mpan 689 . . . 4 (𝑦 ∈ ω → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
1918eqeq1d 2826 . . 3 (𝑦 ∈ ω → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = ∅))
2015, 19syl5ibr 249 . 2 (𝑦 ∈ ω → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
212, 4, 6, 8, 11, 20finds 7600 1 (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ∅c0 4276  Oncon0 6179  suc csuc 6181  (class class class)co 7146  ωcom 7571   +o coa 8091   ·o comu 8092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098  df-omul 8099 This theorem is referenced by:  nnmcom  8244  nnmord  8250  nnmwordi  8253
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