MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfsuc Unicode version

Theorem cantnfsuc 7371
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 6469 . . 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 452 . 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 2796 . . . 4  |-  ( K  e.  om  ->  K  e.  _V )
54adantl 452 . . 3  |-  ( (
ph  /\  K  e.  om )  ->  K  e.  _V )
6 fvex 5539 . . 3  |-  ( H `
 K )  e. 
_V
7 simpl 443 . . . . . . . 8  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  ->  u  =  K )
87fveq2d 5529 . . . . . . 7  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( G `  u
)  =  ( G `
 K ) )
98oveq2d 5874 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( A  ^o  ( G `  u )
)  =  ( A  ^o  ( G `  K ) ) )
108fveq2d 5529 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( F `  ( G `  u )
)  =  ( F `
 ( G `  K ) ) )
119, 10oveq12d 5876 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( ( A  ^o  ( G `  u ) )  .o  ( F `
 ( G `  u ) ) )  =  ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K )
) ) )
12 simpr 447 . . . . 5  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
v  =  ( H `
 K ) )
1311, 12oveq12d 5876 . . . 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 5525 . . . . . . . 8  |-  ( k  =  u  ->  ( G `  k )  =  ( G `  u ) )
1514oveq2d 5874 . . . . . . 7  |-  ( k  =  u  ->  ( A  ^o  ( G `  k ) )  =  ( A  ^o  ( G `  u )
) )
1614fveq2d 5529 . . . . . . 7  |-  ( k  =  u  ->  ( F `  ( G `  k ) )  =  ( F `  ( G `  u )
) )
1715, 16oveq12d 5876 . . . . . 6  |-  ( k  =  u  ->  (
( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  =  ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) ) )
1817oveq1d 5873 . . . . 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 5866 . . . . 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 5926 . . . 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 5883 . . . 4  |-  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) )  e. 
_V
2213, 20, 21ovmpt2a 5978 . . 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 643 . 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 2315 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455    _E cep 4303   Oncon0 4392   suc csuc 4394   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnfle  7372  cantnflt  7373  cantnfp1lem3  7382  cantnflem1d  7390  cantnflem1  7391  cnfcomlem  7402
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-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-om 4657  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-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460
  Copyright terms: Public domain W3C validator