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Theorem onasuc 7190
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7186 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
onasuc  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )

Proof of Theorem onasuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 frsuc 7114 . . . 4  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  suc  x ) `  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B ) ) )
21adantl 466 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  suc  x ) `  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B ) ) )
3 peano2 6715 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
43adantl 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
5 fvres 5886 . . . 4  |-  ( suc 
B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
64, 5syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
7 fvres 5886 . . . . 5  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 B ) )
87adantl 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 B ) )
98fveq2d 5876 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( x  e. 
_V  |->  suc  x ) `  ( ( rec (
( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B ) )  =  ( ( x  e. 
_V  |->  suc  x ) `  ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
) `  B )
) )
102, 6, 93eqtr3d 2516 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( rec ( ( x  e.  _V  |->  suc  x ) ,  A
) `  suc  B )  =  ( ( x  e.  _V  |->  suc  x
) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
11 nnon 6701 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
12 suceloni 6643 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
1311, 12syl 16 . . 3  |-  ( B  e.  om  ->  suc  B  e.  On )
14 oav 7173 . . 3  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
1513, 14sylan2 474 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  suc  B ) )
16 ovex 6320 . . . 4  |-  ( A  +o  B )  e. 
_V
17 suceq 4949 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
18 eqid 2467 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
1916sucex 6641 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
2017, 18, 19fvmpt 5957 . . . 4  |-  ( ( A  +o  B )  e.  _V  ->  (
( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B ) )
2116, 20ax-mp 5 . . 3  |-  ( ( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B )
22 oav 7173 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2311, 22sylan2 474 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2423fveq2d 5876 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( x  e. 
_V  |->  suc  x ) `  ( A  +o  B
) )  =  ( ( x  e.  _V  |->  suc  x ) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
2521, 24syl5eqr 2522 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  ( A  +o  B )  =  ( ( x  e.  _V  |->  suc  x ) `  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) ) )
2610, 15, 253eqtr4d 2518 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511   Oncon0 4884   suc csuc 4886    |` cres 5007   ` cfv 5594  (class class class)co 6295   omcom 6695   reccrdg 7087    +o coa 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-oadd 7146
This theorem is referenced by:  oa1suc  7193  nnasuc  7267
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