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Theorem oelim 6549
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 4471 . . 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 6462 . . . 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 6530 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  B ) )
7 onelon 4433 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oevn0 6530 . . . . . . . . . 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 3942 . . . . 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 2310 . . . 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   U_ciun 3921    e. cmpt 4093   Oncon0 4408   Lim wlim 4409   ` cfv 5271  (class class class)co 5874   reccrdg 6438   1oc1o 6488    .o comu 6493    ^o coe 6494
This theorem is referenced by:  oecl  6552  oe1m  6559  oen0  6600  oeordi  6601  oewordri  6606  oeworde  6607  oelim2  6609  oeoalem  6610  oeoelem  6612  oeeulem  6615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oexp 6501
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