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Theorem onasuc 7176
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7172 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 7100 . . . 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 6701 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
43adantl 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
5 fvres 5866 . . . 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 5866 . . . . 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 5856 . . 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 2490 . 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 6687 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
12 suceloni 6629 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
1311, 12syl 16 . . 3  |-  ( B  e.  om  ->  suc  B  e.  On )
14 oav 7159 . . 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 6305 . . . 4  |-  ( A  +o  B )  e. 
_V
17 suceq 4929 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
18 eqid 2441 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
1916sucex 6627 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
2017, 18, 19fvmpt 5937 . . . 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 7159 . . . . 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 5856 . . 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 2496 . 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 2492 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 1381    e. wcel 1802   _Vcvv 3093    |-> cmpt 4491   Oncon0 4864   suc csuc 4866    |` cres 4987   ` cfv 5574  (class class class)co 6277   omcom 6681   reccrdg 7073    +o coa 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-oadd 7132
This theorem is referenced by:  oa1suc  7179  nnasuc  7253
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