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Theorem om0x 7544
 Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 7542, 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.)
Assertion
Ref Expression
om0x (𝐴 ·𝑜 ∅) = ∅

Proof of Theorem om0x
StepHypRef Expression
1 om0 7542 . . 3 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
21adantr 481 . 2 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
3 fnom 7534 . . . 4 ·𝑜 Fn (On × On)
4 fndm 5948 . . . 4 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
53, 4ax-mp 5 . . 3 dom ·𝑜 = (On × On)
65ndmov 6771 . 2 (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
72, 6pm2.61i 176 1 (𝐴 ·𝑜 ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∅c0 3891   × cxp 5072  dom cdm 5074  Oncon0 5682   Fn wfn 5842  (class class class)co 6604   ·𝑜 comu 7503 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-omul 7510 This theorem is referenced by:  om0r  7564  om1r  7568  omeulem1  7607  nnm0r  7635  nneob  7677  fin1a2lem6  9171
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