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Theorem oelim 6419
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
StepHypRef Expression
1 limelon 4348 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 449 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 520 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6332 . . . 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 710 . . 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 6400 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  B ) )
7 onelon 4310 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oevn0 6400 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  .o  A ) ) ,  1o ) `  x ) )
97, 8sylanl2 635 . . . . . . . . 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 597 . . . . . . . 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 579 . . . . . 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 3824 . . . . 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 2267 . . . 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 704 . . 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 225 . 2  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
173, 16sylanl2 635 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2727   (/)c0 3362   U_ciun 3803    e. cmpt 3974   Oncon0 4285   Lim wlim 4286   ` cfv 4592  (class class class)co 5710   reccrdg 6308   1oc1o 6358    .o comu 6363    ^o coe 6364
This theorem is referenced by:  oecl  6422  oe1m  6429  oen0  6470  oeordi  6471  oewordri  6476  oeworde  6477  oelim2  6479  oeoalem  6480  oeoelem  6482  oeeulem  6485
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-recs 6274  df-rdg 6309  df-1o 6365  df-oexp 6371
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