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Theorem cantnfsuc 8080
Description: The value of the recursive function  H at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfcl.g  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
cantnfcl.f  |-  ( ph  ->  F  e.  S )
cantnfval.h  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
Assertion
Ref Expression
cantnfsuc  |-  ( (
ph  /\  K  e.  om )  ->  ( H `  suc  K )  =  ( ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) )  +o  ( H `  K )
) )
Distinct variable groups:    z, k, B    A, k, z    k, F, z    S, k, z   
k, G, z    k, K, z    ph, k, z
Allowed substitution hints:    H( z, k)

Proof of Theorem cantnfsuc
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfval.h . . . 4  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
21seqomsuc 7114 . . 3  |-  ( K  e.  om  ->  ( H `  suc  K )  =  ( K ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )  +o  z ) ) ( H `  K
) ) )
32adantl 464 . 2  |-  ( (
ph  /\  K  e.  om )  ->  ( H `  suc  K )  =  ( K ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  +o  z ) ) ( H `  K
) ) )
4 elex 3115 . . . 4  |-  ( K  e.  om  ->  K  e.  _V )
54adantl 464 . . 3  |-  ( (
ph  /\  K  e.  om )  ->  K  e.  _V )
6 fvex 5858 . . 3  |-  ( H `
 K )  e. 
_V
7 simpl 455 . . . . . . . 8  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  ->  u  =  K )
87fveq2d 5852 . . . . . . 7  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( G `  u
)  =  ( G `
 K ) )
98oveq2d 6286 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( A  ^o  ( G `  u )
)  =  ( A  ^o  ( G `  K ) ) )
108fveq2d 5852 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( F `  ( G `  u )
)  =  ( F `
 ( G `  K ) ) )
119, 10oveq12d 6288 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( ( A  ^o  ( G `  u ) )  .o  ( F `
 ( G `  u ) ) )  =  ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) ) )
12 simpr 459 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
v  =  ( H `
 K ) )
1311, 12oveq12d 6288 . . . 4  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) )  +o  v
)  =  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) ) )
14 fveq2 5848 . . . . . . . 8  |-  ( k  =  u  ->  ( G `  k )  =  ( G `  u ) )
1514oveq2d 6286 . . . . . . 7  |-  ( k  =  u  ->  ( A  ^o  ( G `  k ) )  =  ( A  ^o  ( G `  u )
) )
1614fveq2d 5852 . . . . . . 7  |-  ( k  =  u  ->  ( F `  ( G `  k ) )  =  ( F `  ( G `  u )
) )
1715, 16oveq12d 6288 . . . . . 6  |-  ( k  =  u  ->  (
( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  =  ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) ) )
1817oveq1d 6285 . . . . 5  |-  ( k  =  u  ->  (
( ( A  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )  +o  z )  =  ( ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) )  +o  z
) )
19 oveq2 6278 . . . . 5  |-  ( z  =  v  ->  (
( ( A  ^o  ( G `  u ) )  .o  ( F `
 ( G `  u ) ) )  +o  z )  =  ( ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) )  +o  v
) )
2018, 19cbvmpt2v 6350 . . . 4  |-  ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  +o  z ) )  =  ( u  e. 
_V ,  v  e. 
_V  |->  ( ( ( A  ^o  ( G `
 u ) )  .o  ( F `  ( G `  u ) ) )  +o  v
) )
21 ovex 6298 . . . 4  |-  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) )  e. 
_V
2213, 20, 21ovmpt2a 6406 . . 3  |-  ( ( K  e.  _V  /\  ( H `  K )  e.  _V )  -> 
( K ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  +o  z ) ) ( H `  K
) )  =  ( ( ( A  ^o  ( G `  K ) )  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) ) )
235, 6, 22sylancl 660 . 2  |-  ( (
ph  /\  K  e.  om )  ->  ( K
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ( H `
 K ) )  =  ( ( ( A  ^o  ( G `
 K ) )  .o  ( F `  ( G `  K ) ) )  +o  ( H `  K )
) )
243, 23eqtrd 2495 1  |-  ( (
ph  /\  K  e.  om )  ->  ( H `  suc  K )  =  ( ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) )  +o  ( H `  K )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783    _E cep 4778   Oncon0 4867   suc csuc 4869   dom cdm 4988   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   omcom 6673   supp csupp 6891  seq𝜔cseqom 7104    +o coa 7119    .o comu 7120    ^o coe 7121  OrdIsocoi 7926   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-seqom 7105
This theorem is referenced by:  cantnfle  8081  cantnflt  8082  cantnfp1lem3  8090  cantnflem1d  8098  cantnflem1  8099  cnfcomlem  8134
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