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Theorem oasuc 6523
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 6437 . . 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 4604 . . 3  |-  ( B  e.  On  ->  suc  B  e.  On )
4 oav 6510 . . 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 5883 . . . 4  |-  ( A  +o  B )  e. 
_V
7 suceq 4457 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
8 eqid 2283 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
96sucex 4602 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
107, 8, 9fvmpt 5602 . . . 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 6510 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
1312fveq2d 5529 . . 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 2329 . 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 2325 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 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   Oncon0 4392   suc csuc 4394   ` cfv 5255  (class class class)co 5858   reccrdg 6422    +o coa 6476
This theorem is referenced by:  oacl  6534  oa0r  6537  oaordi  6544  oawordri  6548  oawordeulem  6552  oalimcl  6558  oaass  6559  oarec  6560  odi  6577  oeoalem  6594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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