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Theorem cantnfsuc 7984
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 7017 . . 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 466 . 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 3081 . . . 4  |-  ( K  e.  om  ->  K  e.  _V )
54adantl 466 . . 3  |-  ( (
ph  /\  K  e.  om )  ->  K  e.  _V )
6 fvex 5804 . . 3  |-  ( H `
 K )  e. 
_V
7 simpl 457 . . . . . . . 8  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  ->  u  =  K )
87fveq2d 5798 . . . . . . 7  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( G `  u
)  =  ( G `
 K ) )
98oveq2d 6211 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( A  ^o  ( G `  u )
)  =  ( A  ^o  ( G `  K ) ) )
108fveq2d 5798 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( F `  ( G `  u )
)  =  ( F `
 ( G `  K ) ) )
119, 10oveq12d 6213 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( ( A  ^o  ( G `  u ) )  .o  ( F `
 ( G `  u ) ) )  =  ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) ) )
12 simpr 461 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
v  =  ( H `
 K ) )
1311, 12oveq12d 6213 . . . 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 5794 . . . . . . . 8  |-  ( k  =  u  ->  ( G `  k )  =  ( G `  u ) )
1514oveq2d 6211 . . . . . . 7  |-  ( k  =  u  ->  ( A  ^o  ( G `  k ) )  =  ( A  ^o  ( G `  u )
) )
1614fveq2d 5798 . . . . . . 7  |-  ( k  =  u  ->  ( F `  ( G `  k ) )  =  ( F `  ( G `  u )
) )
1715, 16oveq12d 6213 . . . . . 6  |-  ( k  =  u  ->  (
( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  =  ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) ) )
1817oveq1d 6210 . . . . 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 6203 . . . . 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 6270 . . . 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 6220 . . . 4  |-  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) )  e. 
_V
2213, 20, 21ovmpt2a 6326 . . 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 662 . 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 2493 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3072   (/)c0 3740    _E cep 4733   Oncon0 4822   suc csuc 4824   dom cdm 4943   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   omcom 6581   supp csupp 6795  seq𝜔cseqom 7007    +o coa 7022    .o comu 7023    ^o coe 7024  OrdIsocoi 7829   CNF ccnf 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-2nd 6683  df-recs 6937  df-rdg 6971  df-seqom 7008
This theorem is referenced by:  cantnfle  7985  cantnflt  7986  cantnfp1lem3  7994  cantnflem1d  8002  cantnflem1  8003  cnfcomlem  8038
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