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Theorem oe0m 6517
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 4445 . . 3  |-  (/)  e.  On
2 oev 6513 . . 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 2283 . . 3  |-  (/)  =  (/)
5 iftrue 3571 . . 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 2331 1  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ifcif 3565    e. cmpt 4077   Oncon0 4392   ` cfv 5255  (class class class)co 5858   reccrdg 6422   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem is referenced by:  oe0m0  6519  oe0m1  6520  cantnflem2  7392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oexp 6485
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