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Theorem oav 6215
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 oafnex 6205 . . 3  |-  ( x  e.  _V  |->  suc  x
)  Fn  _V
21rdgexgg 6143 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
) `  B )  e.  _V )
3 rdgeq2 6137 . . . 4  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  suc  x ) ,  y )  =  rec ( ( x  e.  _V  |->  suc  x
) ,  A ) )
43fveq1d 5307 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  y ) `  z )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 z ) )
5 fveq2 5305 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
6 df-oadd 6185 . . 3  |-  +o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  suc  x
) ,  y ) `
 z ) )
74, 5, 6ovmpt2g 5779 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
)  e.  _V )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
82, 7mpd3an3 1274 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 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    |-> cmpt 3899   Oncon0 4190   suc csuc 4192   ` cfv 5015  (class class class)co 5652   reccrdg 6134    +o coa 6178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-recs 6070  df-irdg 6135  df-oadd 6185
This theorem is referenced by:  oa0  6218  oacl  6221  oav2  6224  oawordi  6230
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