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Theorem onasuc 6495
Description: Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6491 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
StepHypRef Expression
1 frsuc 6417 . . . 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 454 . . 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 4648 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
43adantl 454 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
5 fvres 5475 . . . 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 17 . . 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 5475 . . . . 5  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  suc  x ) ,  A
)  |`  om ) `  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 B ) )
87adantl 454 . . . 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 5462 . . 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 2298 . 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 4634 . . . 4  |-  ( B  e.  om  ->  B  e.  On )
12 suceloni 4576 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
1311, 12syl 17 . . 3  |-  ( B  e.  om  ->  suc  B  e.  On )
14 oav 6478 . . 3  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e.  _V  |->  suc  x
) ,  A ) `
 suc  B )
)
1513, 14sylan2 462 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  suc  B ) )
16 ovex 5817 . . . 4  |-  ( A  +o  B )  e. 
_V
17 suceq 4429 . . . . 5  |-  ( x  =  ( A  +o  B )  ->  suc  x  =  suc  ( A  +o  B ) )
18 eqid 2258 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  =  ( x  e.  _V  |->  suc  x
)
1916sucex 4574 . . . . 5  |-  suc  ( A  +o  B )  e. 
_V
2017, 18, 19fvmpt 5536 . . . 4  |-  ( ( A  +o  B )  e.  _V  ->  (
( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B ) )
2116, 20ax-mp 10 . . 3  |-  ( ( x  e.  _V  |->  suc  x ) `  ( A  +o  B ) )  =  suc  ( A  +o  B )
22 oav 6478 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2311, 22sylan2 462 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  B
)  =  ( rec ( ( x  e. 
_V  |->  suc  x ) ,  A ) `  B
) )
2423fveq2d 5462 . . 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 2304 . 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 2300 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2763    e. cmpt 4051   Oncon0 4364   suc csuc 4366   omcom 4628    |` cres 4663   ` cfv 4673  (class class class)co 5792   reccrdg 6390    +o coa 6444
This theorem is referenced by:  oa1suc  6498  nnasuc  6572
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-oadd 6451
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