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Theorem onasuc 7081
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7077 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 7005 . . . 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 6609 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
43adantl 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
5 fvres 5816 . . . 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 5816 . . . . 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 5806 . . 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 2503 . 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 6595 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
12 suceloni 6537 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
1311, 12syl 16 . . 3  |-  ( B  e.  om  ->  suc  B  e.  On )
14 oav 7064 . . 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 6228 . . . 4  |-  ( A  +o  B )  e. 
_V
17 suceq 4895 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
18 eqid 2454 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
1916sucex 6535 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
2017, 18, 19fvmpt 5886 . . . 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 7064 . . . . 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 5806 . . 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 2509 . 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 2505 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 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461   Oncon0 4830   suc csuc 4832    |` cres 4953   ` cfv 5529  (class class class)co 6203   omcom 6589   reccrdg 6978    +o coa 7030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-oadd 7037
This theorem is referenced by:  oa1suc  7084  nnasuc  7158
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