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Theorem oav 6757
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 6672 . . 3  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  suc  x ) ,  y )  =  rec ( ( x  e.  _V  |->  suc  x
) ,  A ) )
21fveq1d 5732 . 2  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  y ) `  z )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 z ) )
3 fveq2 5730 . 2  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
4 df-oadd 6730 . 2  |-  +o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  suc  x
) ,  y ) `
 z ) )
5 fvex 5744 . 2  |-  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
)  e.  _V
62, 3, 4, 5ovmpt2 6211 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4268   Oncon0 4583   suc csuc 4585   ` cfv 5456  (class class class)co 6083   reccrdg 6669    +o coa 6723
This theorem is referenced by:  oa0  6762  oasuc  6770  onasuc  6774  oalim  6778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-oadd 6730
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