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Theorem oav 7161
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 7078 . . 3  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  suc  x ) ,  y )  =  rec ( ( x  e.  _V  |->  suc  x
) ,  A ) )
21fveq1d 5868 . 2  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  y ) `  z )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 z ) )
3 fveq2 5866 . 2  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
4 df-oadd 7134 . 2  |-  +o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  suc  x
) ,  y ) `
 z ) )
5 fvex 5876 . 2  |-  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
)  e.  _V
62, 3, 4, 5ovmpt2 6422 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   Oncon0 4878   suc csuc 4880   ` cfv 5588  (class class class)co 6284   reccrdg 7075    +o coa 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-recs 7042  df-rdg 7076  df-oadd 7134
This theorem is referenced by:  oa0  7166  oasuc  7174  onasuc  7178  oalim  7182
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