Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  om0 Structured version   Visualization version   GIF version

Theorem om0 8139
 Description: Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. See om0x 8141 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 6232 . . 3 ∅ ∈ On
2 omv 8134 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅))
31, 2mpan2 690 . 2 (𝐴 ∈ On → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅))
4 0ex 5198 . . 3 ∅ ∈ V
54rdg0 8054 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅) = ∅
63, 5syl6eq 2875 1 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3481  ∅c0 4277   ↦ cmpt 5133  Oncon0 6179  ‘cfv 6344  (class class class)co 7150  reccrdg 8042   +o coa 8096   ·o comu 8097 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 7456 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 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  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 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-omul 8104 This theorem is referenced by:  om0x  8141  oesuclem  8147  omcl  8158  om0r  8161  om1  8165  om1r  8166  omwordri  8195  om00  8198  odi  8202  omass  8203  omeulem1  8205  oen0  8209  oeoa  8220  oeoelem  8221  oeeui  8225  nnm0  8228  nnm0r  8233  nneob  8276  cantnfle  9132  cantnfp1  9142  fin1a2lem6  9826
 Copyright terms: Public domain W3C validator