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Theorem oav 7221
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 7138 . . 3  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  suc  x ) ,  y )  =  rec ( ( x  e.  _V  |->  suc  x
) ,  A ) )
21fveq1d 5883 . 2  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  y ) `  z )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 z ) )
3 fveq2 5881 . 2  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
4 df-oadd 7194 . 2  |-  +o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  suc  x
) ,  y ) `
 z ) )
5 fvex 5891 . 2  |-  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
)  e.  _V
62, 3, 4, 5ovmpt2 6446 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 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    |-> cmpt 4484   Oncon0 5442   suc csuc 5444   ` cfv 5601  (class class class)co 6305   reccrdg 7135    +o coa 7187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-oadd 7194
This theorem is referenced by:  oa0  7226  oasuc  7234  onasuc  7238  oalim  7242
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