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Theorem om0x 6514
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6512, 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 6512 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
21adantr 453 . 2  |-  ( ( A  e.  On  /\  (/) 
e.  On )  -> 
( A  .o  (/) )  =  (/) )
3 fnom 6504 . . . 4  |-  .o  Fn  ( On  X.  On )
4 fndm 5309 . . . 4  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
53, 4ax-mp 10 . . 3  |-  dom  .o  =  ( On  X.  On )
65ndmov 5966 . 2  |-  ( -.  ( A  e.  On  /\  (/)  e.  On )  -> 
( A  .o  (/) )  =  (/) )
72, 6pm2.61i 158 1  |-  ( A  .o  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1624    e. wcel 1685   (/)c0 3457   Oncon0 4392    X. cxp 4687   dom cdm 4689    Fn wfn 5217  (class class class)co 5820    .o comu 6473
This theorem is referenced by:  om0r  6534  om1r  6537  omeulem1  6576  nnm0r  6604  nneob  6646  fin1a2lem6  8027
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-omul 6480
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