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Theorem oav2 6074
Description: Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
Assertion
Ref Expression
oav2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem oav2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oafnex 6055 . . 3  |-  ( y  e.  _V  |->  suc  y
)  Fn  _V
2 rdgival 6000 . . 3  |-  ( ( ( y  e.  _V  |->  suc  y )  Fn  _V  /\  A  e.  On  /\  B  e.  On )  ->  ( rec ( ( y  e.  _V  |->  suc  y ) ,  A
) `  B )  =  ( A  u.  U_ x  e.  B  ( ( y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) ) )
31, 2mp3an1 1230 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( y  e.  _V  |->  suc  y ) ,  A
) `  B )  =  ( A  u.  U_ x  e.  B  ( ( y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) ) )
4 oav 6065 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  B
) )
5 onelon 4149 . . . . . 6  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
6 vex 2577 . . . . . . . . . 10  |-  x  e. 
_V
7 oaexg 6059 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  x  e.  _V )  ->  ( A  +o  x
)  e.  _V )
86, 7mpan2 409 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A  +o  x )  e. 
_V )
9 sucexg 4252 . . . . . . . . . 10  |-  ( ( A  +o  x )  e.  _V  ->  suc  ( A  +o  x
)  e.  _V )
108, 9syl 14 . . . . . . . . 9  |-  ( A  e.  On  ->  suc  ( A  +o  x
)  e.  _V )
11 suceq 4167 . . . . . . . . . 10  |-  ( y  =  ( A  +o  x )  ->  suc  y  =  suc  ( A  +o  x ) )
12 eqid 2056 . . . . . . . . . 10  |-  ( y  e.  _V  |->  suc  y
)  =  ( y  e.  _V  |->  suc  y
)
1311, 12fvmptg 5276 . . . . . . . . 9  |-  ( ( ( A  +o  x
)  e.  _V  /\  suc  ( A  +o  x
)  e.  _V )  ->  ( ( y  e. 
_V  |->  suc  y ) `  ( A  +o  x
) )  =  suc  ( A  +o  x
) )
148, 10, 13syl2anc 397 . . . . . . . 8  |-  ( A  e.  On  ->  (
( y  e.  _V  |->  suc  y ) `  ( A  +o  x ) )  =  suc  ( A  +o  x ) )
1514adantr 265 . . . . . . 7  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( ( y  e. 
_V  |->  suc  y ) `  ( A  +o  x
) )  =  suc  ( A  +o  x
) )
16 oav 6065 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  =  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) )
1716fveq2d 5210 . . . . . . 7  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( ( y  e. 
_V  |->  suc  y ) `  ( A  +o  x
) )  =  ( ( y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) )
1815, 17eqtr3d 2090 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  On )  ->  suc  ( A  +o  x )  =  ( ( y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) )
195, 18sylan2 274 . . . . 5  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  suc  ( A  +o  x )  =  ( ( y  e. 
_V  |->  suc  y ) `  ( rec ( ( y  e.  _V  |->  suc  y ) ,  A
) `  x )
) )
2019anassrs 386 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  suc  ( A  +o  x )  =  ( ( y  e. 
_V  |->  suc  y ) `  ( rec ( ( y  e.  _V  |->  suc  y ) ,  A
) `  x )
) )
2120iuneq2dv 3706 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U_ x  e.  B  suc  ( A  +o  x
)  =  U_ x  e.  B  ( (
y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) )
2221uneq2d 3125 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  U_ x  e.  B  suc  ( A  +o  x
) )  =  ( A  u.  U_ x  e.  B  ( (
y  e.  _V  |->  suc  y ) `  ( rec ( ( y  e. 
_V  |->  suc  y ) ,  A ) `  x
) ) ) )
233, 4, 223eqtr4d 2098 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    u. cun 2943   U_ciun 3685    |-> cmpt 3846   Oncon0 4128   suc csuc 4130    Fn wfn 4925   ` cfv 4930  (class class class)co 5540   reccrdg 5987    +o coa 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036
This theorem is referenced by:  oasuc  6075
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