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Theorem oelim 6716
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 4587 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 448 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 519 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6629 . . . 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 708 . . 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 6697 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  B ) )
7 onelon 4549 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oevn0 6697 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) )
97, 8sylanl2 633 . . . . . . . . 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 595 . . . . . . . 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 79 . . . . . . 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 577 . . . . . 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 4058 . . . . 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 2403 . . . 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 702 . . 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 224 . 2  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
173, 16sylanl2 633 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573   U_ciun 4037    e. cmpt 4209   Oncon0 4524   Lim wlim 4525   ` cfv 5396  (class class class)co 6022   reccrdg 6605   1oc1o 6655    .o comu 6660    ^o coe 6661
This theorem is referenced by:  oecl  6719  oe1m  6726  oen0  6767  oeordi  6768  oewordri  6773  oeworde  6774  oelim2  6776  oeoalem  6777  oeoelem  6779  oeeulem  6782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-1o 6662  df-oexp 6668
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