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Theorem oelim 6535
Description: Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oelim  |-  ( ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem oelim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 limelon 4457 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 447 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 518 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6448 . . . 4  |-  ( ( B  e.  On  /\  Lim  B )  ->  ( rec ( ( y  e. 
_V  |->  ( y  .o  A ) ) ,  1o ) `  B
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  ( y  .o  A ) ) ,  1o ) `  x
) )
54ad2antlr 707 . . 3  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  /\  (/)  e.  A )  ->  ( rec (
( y  e.  _V  |->  ( y  .o  A
) ) ,  1o ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) )
6 oevn0 6516 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  B ) )
7 onelon 4419 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oevn0 6516 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) )
97, 8sylanl2 632 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  /\  (/)  e.  A
)  ->  ( A  ^o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) )
109exp42 594 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  On  ->  ( x  e.  B  -> 
( (/)  e.  A  -> 
( A  ^o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  .o  A ) ) ,  1o ) `  x
) ) ) ) )
1110com34 77 . . . . . . 7  |-  ( A  e.  On  ->  ( B  e.  On  ->  (
(/)  e.  A  ->  ( x  e.  B  -> 
( A  ^o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  .o  A ) ) ,  1o ) `  x
) ) ) ) )
1211imp41 576 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A
)  /\  x  e.  B )  ->  ( A  ^o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `
 x ) )
1312iuneq2dv 3928 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  U_ x  e.  B  ( A  ^o  x
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  ( y  .o  A ) ) ,  1o ) `  x
) )
146, 13eqeq12d 2299 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( ( A  ^o  B )  = 
U_ x  e.  B  ( A  ^o  x
)  <->  ( rec (
( y  e.  _V  |->  ( y  .o  A
) ) ,  1o ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) ) )
1514adantlrr 701 . . 3  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  /\  (/)  e.  A )  ->  ( ( A  ^o  B )  = 
U_ x  e.  B  ( A  ^o  x
)  <->  ( rec (
( y  e.  _V  |->  ( y  .o  A
) ) ,  1o ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) ) )
165, 15mpbird 223 . 2  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
173, 16sylanl2 632 1  |-  ( ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   _Vcvv 2790   (/)c0 3457   U_ciun 3907    e. cmpt 4079   Oncon0 4394   Lim wlim 4395   ` cfv 5257  (class class class)co 5860   reccrdg 6424   1oc1o 6474    .o comu 6479    ^o coe 6480
This theorem is referenced by:  oecl  6538  oe1m  6545  oen0  6586  oeordi  6587  oewordri  6592  oeworde  6593  oelim2  6595  oeoalem  6596  oeoelem  6598  oeeulem  6601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-1o 6481  df-oexp 6487
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