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Theorem oevn0 7060
Description: Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oevn0  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem oevn0
StepHypRef Expression
1 on0eln0 4877 . . . . 5  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
2 df-ne 2647 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
31, 2syl6bb 261 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  -.  A  =  (/) ) )
43adantr 465 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  <->  -.  A  =  (/) ) )
5 oev 7059 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  =  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
) )
6 iffalse 3902 . . . . 5  |-  ( -.  A  =  (/)  ->  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
75, 6sylan9eq 2513 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  -.  A  =  (/) )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B ) )
87ex 434 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  =  (/)  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) ) )
94, 8sylbid 215 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  ->  ( A  ^o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) ) )
109imp 429 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   _Vcvv 3072    \ cdif 3428   (/)c0 3740   ifcif 3894    |-> cmpt 4453   Oncon0 4822   ` cfv 5521  (class class class)co 6195   reccrdg 6970   1oc1o 7018    .o comu 7023    ^o coe 7024
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
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 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-recs 6937  df-rdg 6971  df-1o 7025  df-oexp 7031
This theorem is referenced by:  oe0  7067  oev2  7068  oesuclem  7070  oelim  7079
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