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Theorem oalim 6767
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 4636 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 448 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 519 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6682 . . . 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 453 . . 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 6746 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
) )
7 onelon 4598 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oav 6746 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) )
97, 8sylan2 461 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  +o  x )  =  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
109anassrs 630 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  +o  x )  =  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
1110iuneq2dv 4106 . . . . 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 2449 . . . 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 698 . . 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 224 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  +o  B )  =  U_ x  e.  B  ( A  +o  x ) )
153, 14sylan2 461 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   U_ciun 4085    e. cmpt 4258   Oncon0 4573   Lim wlim 4574   suc csuc 4575   ` cfv 5445  (class class class)co 6072   reccrdg 6658    +o coa 6712
This theorem is referenced by:  oacl  6770  oa0r  6773  oaordi  6780  oawordri  6784  oawordeulem  6788  oalimcl  6794  oaass  6795  oarec  6796  odi  6813  oeoalem  6830  oaabslem  6877  oaabs2  6879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-oadd 6719
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