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Theorem cantnfvalOLD 8117
Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfval 8087 as of 28-Jun-2019. (New usage 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
cantnfvalOLD  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Distinct variable groups:    z, k, B    A, k, z    k, F, z    S, k, z   
k, G, z    ph, k,
z
Allowed substitution hints:    H( z, k)

Proof of Theorem cantnfvalOLD
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
2 cantnfsOLD.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.3 . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffvalOLD 8082 . . 3  |-  ( ph  ->  ( A CNF  B )  =  ( f  e. 
{ g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin } 
|->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
54fveq1d 5868 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( ( f  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) `  F
) )
6 cantnfvalOLD.4 . . . 4  |-  ( ph  ->  F  e.  S )
7 cantnfsOLD.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
81, 2, 3cantnfdmOLD 8083 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
97, 8syl5eq 2520 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
106, 9eleqtrd 2557 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
11 vex 3116 . . . . . . . 8  |-  f  e. 
_V
1211cnvex 6731 . . . . . . 7  |-  `' f  e.  _V
13 imaexg 6721 . . . . . . 7  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
14 eqid 2467 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
1514oiexg 7960 . . . . . . 7  |-  ( ( `' f " ( _V  \  1o ) )  e.  _V  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V )
1612, 13, 15mp2b 10 . . . . . 6  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V
1716a1i 11 . . . . 5  |-  ( f  =  F  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V )
18 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  = OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) ) )
19 simpl 457 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  f  =  F )
2019cnveqd 5178 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  `' f  =  `' F )
2120imaeq1d 5336 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( `' f
" ( _V  \  1o ) )  =  ( `' F " ( _V 
\  1o ) ) )
22 oieq2 7938 . . . . . . . . . . . . . . . . 17  |-  ( ( `' f " ( _V  \  1o ) )  =  ( `' F " ( _V  \  1o ) )  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )
2321, 22syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  -> OrdIso (  _E  ,  ( `' f " ( _V  \  1o ) ) )  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) )
2418, 23eqtrd 2508 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) )
25 cantnfvalOLD.3 . . . . . . . . . . . . . . 15  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2624, 25syl6eqr 2526 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  =  G )
2726fveq1d 5868 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2827oveq2d 6300 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
2919, 27fveq12d 5872 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
3028, 29oveq12d 6302 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  =  ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) ) )
3130oveq1d 6299 . . . . . . . . . 10  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z )  =  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )  +o  z ) )
32313ad2ant1 1017 . . . . . . . . 9  |-  ( ( ( f  =  F  /\  h  = OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) ) )  /\  k  e.  _V  /\  z  e.  _V )  ->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
)  =  ( ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) )  +o  z ) )
3332mpt2eq3dva 6345 . . . . . . . 8  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) )  =  ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) )
34 eqid 2467 . . . . . . . 8  |-  (/)  =  (/)
35 seqomeq12 7119 . . . . . . . 8  |-  ( ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) )  =  ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `
 ( G `  k ) ) )  +o  z ) )  /\  (/)  =  (/) )  -> seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( G `
 k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
) ) ,  (/) ) )
3633, 34, 35sylancl 662 . . . . . . 7  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  -> seq𝜔
( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( G `
 k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
) ) ,  (/) ) )
37 cantnfvalOLD.5 . . . . . . 7  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
3836, 37syl6eqr 2526 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  -> seq𝜔
( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) )  =  H
)
3926dmeqd 5205 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  dom  h  =  dom  G )
4038, 39fveq12d 5872 . . . . 5  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  =  ( H `
 dom  G )
)
4117, 40csbied 3462 . . . 4  |-  ( f  =  F  ->  [_OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  =  ( H `
 dom  G )
)
42 eqid 2467 . . . 4  |-  ( f  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  ( f  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )
43 fvex 5876 . . . 4  |-  ( H `
 dom  G )  e.  _V
4441, 42, 43fvmpt 5950 . . 3  |-  ( F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  ->  ( ( f  e.  {
g  e.  ( A  ^m  B )  |  ( `' g "
( _V  \  1o ) )  e.  Fin } 
|->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) `  F
)  =  ( H `
 dom  G )
)
4510, 44syl 16 . 2  |-  ( ph  ->  ( ( f  e. 
{ g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin } 
|->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) `  F
)  =  ( H `
 dom  G )
)
465, 45eqtrd 2508 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   [_csb 3435    \ cdif 3473   (/)c0 3785    |-> cmpt 4505    _E cep 4789   Oncon0 4878   `'ccnv 4998   dom cdm 4999   "cima 5002   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286  seq𝜔cseqom 7112   1oc1o 7123    +o coa 7127    .o comu 7128    ^o coe 7129    ^m cmap 7420   Fincfn 7516  OrdIsocoi 7934   CNF ccnf 8078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-seqom 7113  df-1o 7130  df-map 7422  df-fsupp 7830  df-oi 7935  df-cnf 8079
This theorem is referenced by:  cantnfval2OLD  8118  cantnfleOLD  8120  cantnflt2OLD  8122  cantnfp1lem3OLD  8125  cantnflem1OLD  8131  cantnfOLD  8134  cnfcom2OLD  8154
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