MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oalim Unicode version

Theorem oalim 6417
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
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  |->  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 6396 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
) )
7 onelon 4310 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 oav 6396 . . . . . . . 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 632 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  +o  x )  =  ( rec ( ( y  e.  _V  |->  suc  y
) ,  A ) `
 x ) )
1110iuneq2dv 3824 . . . . 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 2267 . . . 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 700 . . 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 1619    e. wcel 1621   _Vcvv 2727   U_ciun 3803    e. cmpt 3974   Oncon0 4285   Lim wlim 4286   suc csuc 4287   ` cfv 4592  (class class class)co 5710   reccrdg 6308    +o coa 6362
This theorem is referenced by:  oacl  6420  oa0r  6423  oaordi  6430  oawordri  6434  oawordeulem  6438  oalimcl  6444  oaass  6445  oarec  6446  odi  6463  oeoalem  6480  oaabslem  6527  oaabs2  6529
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-oadd 6369
  Copyright terms: Public domain W3C validator