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Theorem cantnfval 7612
Description: The value of the Cantor normal form function. (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
cantnfval  |-  ( 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 cantnfval
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
2 cantnfs.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffval 7607 . . 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 5721 . 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 cantnfval.4 . . . 4  |-  ( ph  ->  F  e.  S )
7 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
81, 2, 3cantnfdm 7608 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
97, 8syl5eq 2479 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
106, 9eleqtrd 2511 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
11 vex 2951 . . . . . . . 8  |-  f  e. 
_V
1211cnvex 5397 . . . . . . 7  |-  `' f  e.  _V
13 imaexg 5208 . . . . . . 7  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
14 eqid 2435 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
1514oiexg 7493 . . . . . . 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 448 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  = OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) ) )
19 simpl 444 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  f  =  F )
2019cnveqd 5039 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  `' f  =  `' F )
2120imaeq1d 5193 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( `' f
" ( _V  \  1o ) )  =  ( `' F " ( _V 
\  1o ) ) )
22 oieq2 7471 . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) )
25 cantnfval.3 . . . . . . . . . . . . . . 15  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2624, 25syl6eqr 2485 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  =  G )
2726fveq1d 5721 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2827oveq2d 6088 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
2919, 27fveq12d 5725 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
3028, 29oveq12d 6090 . . . . . . . . . . 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 6087 . . . . . . . . . 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 978 . . . . . . . . 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 6129 . . . . . . . 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 2435 . . . . . . . 8  |-  (/)  =  (/)
35 seqomeq12 6702 . . . . . . . 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 644 . . . . . . 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 cantnfval.5 . . . . . . 7  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
3836, 37syl6eqr 2485 . . . . . 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 5063 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  dom  h  =  dom  G )
4038, 39fveq12d 5725 . . . . 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 3285 . . . 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 2435 . . . 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 5733 . . . 4  |-  ( H `
 dom  G )  e.  _V
4441, 42, 43fvmpt 5797 . . 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 2467 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948   [_csb 3243    \ cdif 3309   (/)c0 3620    e. cmpt 4258    _E cep 4484   Oncon0 4573   `'ccnv 4868   dom cdm 4869   "cima 4872   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074  seq𝜔cseqom 6695   1oc1o 6708    +o coa 6712    .o comu 6713    ^o coe 6714    ^m cmap 7009   Fincfn 7100  OrdIsocoi 7467   CNF ccnf 7605
This theorem is referenced by:  cantnfval2  7613  cantnfle  7615  cantnflt2  7617  cantnff  7618  cantnf0  7619  cantnfp1lem3  7625  cantnflem1  7634  cantnf  7638  cnfcom2  7648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-recs 6624  df-rdg 6659  df-seqom 6696  df-oi 7468  df-cnf 7606
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