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Theorem om0x 8133
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8131, this version works whether or not 𝐴 is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
om0x (𝐴 ·o ∅) = ∅

Proof of Theorem om0x
StepHypRef Expression
1 om0 8131 . . 3 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
21adantr 481 . 2 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
3 fnom 8123 . . . 4 ·o Fn (On × On)
4 fndm 6448 . . . 4 ( ·o Fn (On × On) → dom ·o = (On × On))
53, 4ax-mp 5 . . 3 dom ·o = (On × On)
65ndmov 7321 . 2 (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
72, 6pm2.61i 183 1 (𝐴 ·o ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  c0 4288   × cxp 5546  dom cdm 5548  Oncon0 6184   Fn wfn 6343  (class class class)co 7145   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-omul 8096
This theorem is referenced by: (None)
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