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Theorem oalim 6779
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
Allowed substitution hint:    C( x)

Proof of Theorem oalim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 limelon 4647 . . 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 6694 . . . 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 6758 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
) )
7 onelon 4609 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oav 6758 . . . . . . . 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 4116 . . . . 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 2452 . . . 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 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   U_ciun 4095    e. cmpt 4269   Oncon0 4584   Lim wlim 4585   suc csuc 4586   ` cfv 5457  (class class class)co 6084   reccrdg 6670    +o coa 6724
This theorem is referenced by:  oacl  6782  oa0r  6785  oaordi  6792  oawordri  6796  oawordeulem  6800  oalimcl  6806  oaass  6807  oarec  6808  odi  6825  oeoalem  6842  oaabslem  6889  oaabs2  6891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-oadd 6731
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