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Theorem om0 7831
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
om0 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)

Proof of Theorem om0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0elon 5988 . . 3 ∅ ∈ On
2 omv 7826 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅))
31, 2mpan2 674 . 2 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅))
4 0ex 4981 . . 3 ∅ ∈ V
54rdg0 7750 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅) = ∅
63, 5syl6eq 2855 1 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2158  Vcvv 3390  c0 4113  cmpt 4919  Oncon0 5933  cfv 6098  (class class class)co 6871  reccrdg 7738   +𝑜 coa 7790   ·𝑜 comu 7791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
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 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-omul 7798
This theorem is referenced by:  om0x  7833  oesuclem  7839  omcl  7850  om1  7856  omwordri  7886  om00  7889  odi  7893  omass  7894  oen0  7900  oeoa  7911  oeoelem  7912  oeeui  7916  nnm0  7919  cantnfle  8812  cantnfp1  8822
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