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Theorem om0x 7753
 Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 7751, 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.) TODO: This theorem is used in 6 proofs! It should be marked with "New usage is discouraged.", and it should be replaced by om0 7751 in the 6 proofs.
Assertion
Ref Expression
om0x (𝐴 ·𝑜 ∅) = ∅

Proof of Theorem om0x
StepHypRef Expression
1 om0 7751 . . 3 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
21adantr 466 . 2 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
3 fnom 7743 . . . 4 ·𝑜 Fn (On × On)
4 fndm 6130 . . . 4 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
53, 4ax-mp 5 . . 3 dom ·𝑜 = (On × On)
65ndmov 6965 . 2 (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
72, 6pm2.61i 176 1 (𝐴 ·𝑜 ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∅c0 4063   × cxp 5247  dom cdm 5249  Oncon0 5866   Fn wfn 6026  (class class class)co 6793   ·𝑜 comu 7711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-omul 7718 This theorem is referenced by:  om0r  7773  om1r  7777  omeulem1  7816  nnm0r  7844  nneob  7886  fin1a2lem6  9429
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