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Theorem om0 8555
Description: Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. See om0x 8557 for a way to remove the antecedent 𝐴 ∈ On. (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 6438 . . 3 ∅ ∈ On
2 omv 8550 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅))
31, 2mpan2 691 . 2 (𝐴 ∈ On → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅))
4 0ex 5307 . . 3 ∅ ∈ V
54rdg0 8461 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅) = ∅
63, 5eqtrdi 2793 1 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cmpt 5225  Oncon0 6384  cfv 6561  (class class class)co 7431  reccrdg 8449   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-omul 8511
This theorem is referenced by:  om0x  8557  oesuclem  8563  omcl  8574  om0r  8577  om1  8580  om1r  8581  omwordri  8610  om00  8613  odi  8617  omass  8618  omeulem1  8620  oen0  8624  oeoa  8635  oeoelem  8636  oeeui  8640  nnm0  8643  nnm0r  8648  nneob  8694  cantnfle  9711  cantnfp1  9721  fin1a2lem6  10445  onexlimgt  43255  om0suclim  43289  oaabsb  43307  dflim5  43342  onmcl  43344  omcl3g  43347
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