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Theorem onasuc 6731
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6727 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 6653 . . . 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 453 . . 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 4824 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
43adantl 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
5 fvres 5704 . . . 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 5704 . . . . 5  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 B ) )
87adantl 453 . . . 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 5691 . . 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 2444 . 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 4810 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
12 suceloni 4752 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
1311, 12syl 16 . . 3  |-  ( B  e.  om  ->  suc  B  e.  On )
14 oav 6714 . . 3  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
1513, 14sylan2 461 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  suc  B ) )
16 ovex 6065 . . . 4  |-  ( A  +o  B )  e. 
_V
17 suceq 4606 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
18 eqid 2404 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
1916sucex 4750 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
2017, 18, 19fvmpt 5765 . . . 4  |-  ( ( A  +o  B )  e.  _V  ->  (
( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B ) )
2116, 20ax-mp 8 . . 3  |-  ( ( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B )
22 oav 6714 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2311, 22sylan2 461 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2423fveq2d 5691 . . 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 2450 . 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 2446 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226   Oncon0 4541   suc csuc 4543   omcom 4804    |` cres 4839   ` cfv 5413  (class class class)co 6040   reccrdg 6626    +o coa 6680
This theorem is referenced by:  oa1suc  6734  nnasuc  6808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-oadd 6687
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