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Theorem oelim 6529
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 4454 . . 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 6442 . . . 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 6510 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  B ) )
7 onelon 4416 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oevn0 6510 . . . . . . . . . 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 3927 . . . . 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 2298 . . . 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 1623    e. wcel 1685   _Vcvv 2789   (/)c0 3456   U_ciun 3906    e. cmpt 4078   Oncon0 4391   Lim wlim 4392   ` cfv 5221  (class class class)co 5820   reccrdg 6418   1oc1o 6468    .o comu 6473    ^o coe 6474
This theorem is referenced by:  oecl  6532  oe1m  6539  oen0  6580  oeordi  6581  oewordri  6586  oeworde  6587  oelim2  6589  oeoalem  6590  oeoelem  6592  oeeulem  6595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-1o 6475  df-oexp 6481
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