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Theorem oav 6510
Description: Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oav  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem oav
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgeq2 6425 . . 3  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  suc  x ) ,  y )  =  rec ( ( x  e.  _V  |->  suc  x
) ,  A ) )
21fveq1d 5527 . 2  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  y ) `  z )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 z ) )
3 fveq2 5525 . 2  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
4 df-oadd 6483 . 2  |-  +o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  suc  x
) ,  y ) `
 z ) )
5 fvex 5539 . 2  |-  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
)  e.  _V
62, 3, 4, 5ovmpt2 5983 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   Oncon0 4392   suc csuc 4394   ` cfv 5255  (class class class)co 5858   reccrdg 6422    +o coa 6476
This theorem is referenced by:  oa0  6515  oasuc  6523  onasuc  6527  oalim  6531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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