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Theorem oe0m 6559
Description: Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oe0m  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )

Proof of Theorem oe0m
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0elon 4482 . . 3  |-  (/)  e.  On
2 oev 6555 . . 3  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/) 
^o  A )  =  if ( (/)  =  (/) ,  ( 1o  \  A
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  (/) ) ) ,  1o ) `  A )
) )
31, 2mpan 651 . 2  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  if ( (/)  =  (/) ,  ( 1o  \  A
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  (/) ) ) ,  1o ) `  A )
) )
4 eqid 2316 . . 3  |-  (/)  =  (/)
5 iftrue 3605 . . 3  |-  ( (/)  =  (/)  ->  if ( (/)  =  (/) ,  ( 1o 
\  A ) ,  ( rec ( ( x  e.  _V  |->  ( x  .o  (/) ) ) ,  1o ) `  A ) )  =  ( 1o  \  A
) )
64, 5ax-mp 8 . 2  |-  if (
(/)  =  (/) ,  ( 1o  \  A ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  (/) ) ) ,  1o ) `  A ) )  =  ( 1o  \  A
)
73, 6syl6eq 2364 1  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183   (/)c0 3489   ifcif 3599    e. cmpt 4114   Oncon0 4429   ` cfv 5292  (class class class)co 5900   reccrdg 6464   1oc1o 6514    .o comu 6519    ^o coe 6520
This theorem is referenced by:  oe0m0  6561  oe0m1  6562  cantnflem2  7437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-1o 6521  df-oexp 6527
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