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Theorem oalim 6526
Description: Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oalim  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  +o  B )  =  U_ x  e.  B  ( A  +o  x ) )
Distinct variable groups:    x, A    x, B
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    C( x)

Proof of Theorem oalim
StepHypRef Expression
1 limelon 4454 . . 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 6441 . . . 4  |-  ( ( B  e.  On  /\  Lim  B )  ->  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) )
54adantl 454 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( rec (
( y  e.  _V  |->  suc  y ) ,  A
) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
6 oav 6505 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
) )
7 onelon 4416 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oav 6505 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) )
97, 8sylan2 462 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  +o  x )  =  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
109anassrs 631 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  +o  x )  =  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
1110iuneq2dv 3927 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U_ x  e.  B  ( A  +o  x
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) )
126, 11eqeq12d 2298 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  U_ x  e.  B  ( A  +o  x )  <->  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) )
1312adantrr 699 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( ( A  +o  B )  = 
U_ x  e.  B  ( A  +o  x
)  <->  ( rec (
( y  e.  _V  |->  suc  y ) ,  A
) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) ) )
145, 13mpbird 225 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  +o  B )  =  U_ x  e.  B  ( A  +o  x ) )
153, 14sylan2 462 1  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( 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 1624    e. wcel 1685   _Vcvv 2789   U_ciun 3906    e. cmpt 4078   Oncon0 4391   Lim wlim 4392   suc csuc 4393   ` cfv 5221  (class class class)co 5819   reccrdg 6417    +o coa 6471
This theorem is referenced by:  oacl  6529  oa0r  6532  oaordi  6539  oawordri  6543  oawordeulem  6547  oalimcl  6553  oaass  6554  oarec  6555  odi  6572  oeoalem  6589  oaabslem  6636  oaabs2  6638
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  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 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-oadd 6478
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