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Theorem cantnfvalOLD 7911
Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfval 7881 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 2443 . . . 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 7876 . . 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 5698 . 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 7877 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
97, 8syl5eq 2487 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
106, 9eleqtrd 2519 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
11 vex 2980 . . . . . . . 8  |-  f  e. 
_V
1211cnvex 6530 . . . . . . 7  |-  `' f  e.  _V
13 imaexg 6520 . . . . . . 7  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
14 eqid 2443 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
1514oiexg 7754 . . . . . . 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 5020 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  `' f  =  `' F )
2120imaeq1d 5173 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( `' f
" ( _V  \  1o ) )  =  ( `' F " ( _V 
\  1o ) ) )
22 oieq2 7732 . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  =  G )
2726fveq1d 5698 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2827oveq2d 6112 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
2919, 27fveq12d 5702 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
3028, 29oveq12d 6114 . . . . . . . . . . 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 6111 . . . . . . . . . 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 1009 . . . . . . . . 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 6155 . . . . . . . 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 2443 . . . . . . . 8  |-  (/)  =  (/)
35 seqomeq12 6914 . . . . . . . 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 2493 . . . . . 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 5047 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  dom  h  =  dom  G )
4038, 39fveq12d 5702 . . . . 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 3319 . . . 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 2443 . . . 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 5706 . . . 4  |-  ( H `
 dom  G )  e.  _V
4441, 42, 43fvmpt 5779 . . 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 2475 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977   [_csb 3293    \ cdif 3330   (/)c0 3642    e. cmpt 4355    _E cep 4635   Oncon0 4724   `'ccnv 4844   dom cdm 4845   "cima 4848   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098  seq𝜔cseqom 6907   1oc1o 6918    +o coa 6922    .o comu 6923    ^o coe 6924    ^m cmap 7219   Fincfn 7315  OrdIsocoi 7728   CNF ccnf 7872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-map 7221  df-fsupp 7626  df-oi 7729  df-cnf 7873
This theorem is referenced by:  cantnfval2OLD  7912  cantnfleOLD  7914  cantnflt2OLD  7916  cantnfp1lem3OLD  7919  cantnflem1OLD  7925  cantnfOLD  7928  cnfcom2OLD  7948
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