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Theorem oasuc 6610
Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oasuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )

Proof of Theorem oasuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rdgsuc 6524 . . 3  |-  ( B  e.  On  ->  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  suc  B )  =  ( ( x  e.  _V  |->  suc  x ) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
21adantl 452 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
) `  suc  B )  =  ( ( x  e.  _V  |->  suc  x
) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
3 suceloni 4686 . . 3  |-  ( B  e.  On  ->  suc  B  e.  On )
4 oav 6597 . . 3  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
53, 4sylan2 460 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  suc  B ) )
6 ovex 5970 . . . 4  |-  ( A  +o  B )  e. 
_V
7 suceq 4539 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
8 eqid 2358 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
96sucex 4684 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
107, 8, 9fvmpt 5685 . . . 4  |-  ( ( A  +o  B )  e.  _V  ->  (
( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B ) )
116, 10ax-mp 8 . . 3  |-  ( ( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B )
12 oav 6597 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
1312fveq2d 5612 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e. 
_V  |->  suc  x ) `  ( A  +o  B
) )  =  ( ( x  e.  _V  |->  suc  x ) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
1411, 13syl5eqr 2404 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  suc  ( A  +o  B )  =  ( ( x  e.  _V  |->  suc  x ) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
152, 5, 143eqtr4d 2400 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    e. cmpt 4158   Oncon0 4474   suc csuc 4476   ` cfv 5337  (class class class)co 5945   reccrdg 6509    +o coa 6563
This theorem is referenced by:  oacl  6621  oa0r  6624  oaordi  6631  oawordri  6635  oawordeulem  6639  oalimcl  6645  oaass  6646  oarec  6647  odi  6664  oeoalem  6681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-oadd 6570
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