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Theorem om0x 6472
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6470, 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 6470 . . 3  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
21adantr 453 . 2  |-  ( ( A  e.  On  /\  (/) 
e.  On )  -> 
( A  .o  (/) )  =  (/) )
3 fnom 6462 . . . 4  |-  .o  Fn  ( On  X.  On )
4 fndm 5267 . . . 4  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
53, 4ax-mp 10 . . 3  |-  dom  .o  =  ( On  X.  On )
65ndmov 5924 . 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 1619    e. wcel 1621   (/)c0 3416   Oncon0 4350    X. cxp 4645   dom cdm 4647    Fn wfn 4654  (class class class)co 5778    .o comu 6431
This theorem is referenced by:  om0r  6492  om1r  6495  omeulem1  6534  nnm0r  6562  nneob  6604  fin1a2lem6  7985
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-recs 6342  df-rdg 6377  df-omul 6438
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