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Theorem cantnfsucOLD 8032
Description: The value of the recursive function  H at a successor. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfsuc 8002 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
cantnfvalOLD.3  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cantnfvalOLD.4  |-  ( ph  ->  F  e.  S )
cantnfvalOLD.5  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
Assertion
Ref Expression
cantnfsucOLD  |-  ( (
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 cantnfsucOLD
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfvalOLD.5 . . . 4  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
21seqomsuc 7040 . . 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 3043 . . . 4  |-  ( K  e.  om  ->  K  e.  _V )
54adantl 464 . . 3  |-  ( (
ph  /\  K  e.  om )  ->  K  e.  _V )
6 fvex 5784 . . 3  |-  ( H `
 K )  e. 
_V
7 simpl 455 . . . . . . . 8  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  ->  u  =  K )
87fveq2d 5778 . . . . . . 7  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( G `  u
)  =  ( G `
 K ) )
98oveq2d 6212 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( A  ^o  ( G `  u )
)  =  ( A  ^o  ( G `  K ) ) )
108fveq2d 5778 . . . . . 6  |-  ( ( u  =  K  /\  v  =  ( H `  K ) )  -> 
( F `  ( G `  u )
)  =  ( F `
 ( G `  K ) ) )
119, 10oveq12d 6214 . . . . 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 6214 . . . 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 5774 . . . . . . . 8  |-  ( k  =  u  ->  ( G `  k )  =  ( G `  u ) )
1514oveq2d 6212 . . . . . . 7  |-  ( k  =  u  ->  ( A  ^o  ( G `  k ) )  =  ( A  ^o  ( G `  u )
) )
1614fveq2d 5778 . . . . . . 7  |-  ( k  =  u  ->  ( F `  ( G `  k ) )  =  ( F `  ( G `  u )
) )
1715, 16oveq12d 6214 . . . . . 6  |-  ( k  =  u  ->  (
( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  =  ( ( A  ^o  ( G `  u ) )  .o  ( F `  ( G `  u )
) ) )
1817oveq1d 6211 . . . . 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 6204 . . . . 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 6276 . . . 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 6224 . . . 4  |-  ( ( ( A  ^o  ( G `  K )
)  .o  ( F `
 ( G `  K ) ) )  +o  ( H `  K ) )  e. 
_V
2213, 20, 21ovmpt2a 6332 . . 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 2423 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 1399    e. wcel 1826   _Vcvv 3034    \ cdif 3386   (/)c0 3711    _E cep 4703   Oncon0 4792   suc csuc 4794   `'ccnv 4912   dom cdm 4913   "cima 4916   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   omcom 6599  seq𝜔cseqom 7030   1oc1o 7041    +o coa 7045    .o comu 7046    ^o coe 7047  OrdIsocoi 7849   CNF ccnf 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-seqom 7031
This theorem is referenced by:  cantnfleOLD  8033  cantnfltOLD  8034  cantnfp1lem3OLD  8038  cantnflem1dOLD  8043  cantnflem1OLD  8044  cnfcomlemOLD  8064
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