MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0m Structured version   Unicode version

Theorem oe0m 7044
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 4856 . . 3  |-  (/)  e.  On
2 oev 7040 . . 3  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/) 
^o  A )  =  if ( (/)  =  (/) ,  ( 1o  \  A
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  (/) ) ) ,  1o ) `  A )
) )
31, 2mpan 670 . 2  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  if ( (/)  =  (/) ,  ( 1o  \  A
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  (/) ) ) ,  1o ) `  A )
) )
4 eqid 2450 . . 3  |-  (/)  =  (/)
54iftruei 3882 . 2  |-  if (
(/)  =  (/) ,  ( 1o  \  A ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  (/) ) ) ,  1o ) `  A ) )  =  ( 1o  \  A
)
63, 5syl6eq 2506 1  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   _Vcvv 3054    \ cdif 3409   (/)c0 3721   ifcif 3875    |-> cmpt 4434   Oncon0 4803   ` cfv 5502  (class class class)co 6176   reccrdg 6951   1oc1o 6999    .o comu 7004    ^o coe 7005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-recs 6918  df-rdg 6952  df-1o 7006  df-oexp 7012
This theorem is referenced by:  oe0m0  7046  oe0m1  7047  cantnflem2  7985
  Copyright terms: Public domain W3C validator