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Theorem om0x 6692
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6690, 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 6690 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
21adantr 452 . 2  |-  ( ( A  e.  On  /\  (/) 
e.  On )  -> 
( A  .o  (/) )  =  (/) )
3 fnom 6682 . . . 4  |-  .o  Fn  ( On  X.  On )
4 fndm 5477 . . . 4  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
53, 4ax-mp 8 . . 3  |-  dom  .o  =  ( On  X.  On )
65ndmov 6163 . 2  |-  ( -.  ( A  e.  On  /\  (/)  e.  On )  -> 
( A  .o  (/) )  =  (/) )
72, 6pm2.61i 158 1  |-  ( A  .o  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   (/)c0 3564   Oncon0 4515    X. cxp 4809   dom cdm 4811    Fn wfn 5382  (class class class)co 6013    .o comu 6651
This theorem is referenced by:  om0r  6712  om1r  6715  omeulem1  6754  nnm0r  6782  nneob  6824  fin1a2lem6  8211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-omul 6658
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