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Theorem cantnfsuc 7627
Description: The value of the recursive function  H at a successor. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfval.3  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cantnfval.4  |-  ( ph  ->  F  e.  S )
cantnfval.5  |-  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.5 . . . 4  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
21seqomsuc 6716 . . 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 454 . 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 2966 . . . 4  |-  ( K  e.  om  ->  K  e.  _V )
54adantl 454 . . 3  |-  ( (
ph  /\  K  e.  om )  ->  K  e.  _V )
6 fvex 5744 . . 3  |-  ( H `
 K )  e. 
_V
7 simpl 445 . . . . . . . 8  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  ->  u  =  K )
87fveq2d 5734 . . . . . . 7  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( G `  u
)  =  ( G `
 K ) )
98oveq2d 6099 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( A  ^o  ( G `  u )
)  =  ( A  ^o  ( G `  K ) ) )
108fveq2d 5734 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( F `  ( G `  u )
)  =  ( F `
 ( G `  K ) ) )
119, 10oveq12d 6101 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( ( A  ^o  ( G `  u ) )  .o  ( F `
 ( G `  u ) ) )  =  ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) ) )
12 simpr 449 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
v  =  ( H `
 K ) )
1311, 12oveq12d 6101 . . . 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 5730 . . . . . . . 8  |-  ( k  =  u  ->  ( G `  k )  =  ( G `  u ) )
1514oveq2d 6099 . . . . . . 7  |-  ( k  =  u  ->  ( A  ^o  ( G `  k ) )  =  ( A  ^o  ( G `  u )
) )
1614fveq2d 5734 . . . . . . 7  |-  ( k  =  u  ->  ( F `  ( G `  k ) )  =  ( F `  ( G `  u )
) )
1715, 16oveq12d 6101 . . . . . 6  |-  ( k  =  u  ->  (
( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  =  ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) ) )
1817oveq1d 6098 . . . . 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 6091 . . . . 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 6154 . . . 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 6108 . . . 4  |-  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) )  e. 
_V
2213, 20, 21ovmpt2a 6206 . . 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 645 . 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 2470 1  |-  ( (
ph  /\  K  e.  om )  ->  ( H `  suc  K )  =  ( ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) )  +o  ( H `  K )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   (/)c0 3630    _E cep 4494   Oncon0 4583   suc csuc 4585   omcom 4847   `'ccnv 4879   dom cdm 4880   "cima 4883   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085  seq𝜔cseqom 6706   1oc1o 6719    +o coa 6723    .o comu 6724    ^o coe 6725  OrdIsocoi 7480   CNF ccnf 7618
This theorem is referenced by:  cantnfle  7628  cantnflt  7629  cantnfp1lem3  7638  cantnflem1d  7646  cantnflem1  7647  cnfcomlem  7658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-recs 6635  df-rdg 6670  df-seqom 6707
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