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Theorem cantnfval 7549
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 2380 . . . 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 7544 . . 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 5663 . 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 7545 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
97, 8syl5eq 2424 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
106, 9eleqtrd 2456 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
11 vex 2895 . . . . . . . 8  |-  f  e. 
_V
1211cnvex 5339 . . . . . . 7  |-  `' f  e.  _V
13 imaexg 5150 . . . . . . 7  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
14 eqid 2380 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
1514oiexg 7430 . . . . . . 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 4981 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  `' f  =  `' F )
2120imaeq1d 5135 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( `' f
" ( _V  \  1o ) )  =  ( `' F " ( _V 
\  1o ) ) )
22 oieq2 7408 . . . . . . . . . . . . . . . . 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 2412 . . . . . . . . . . . . . . 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 2430 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  h  =  G )
2726fveq1d 5663 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2827oveq2d 6029 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
2919, 27fveq12d 5667 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
3028, 29oveq12d 6031 . . . . . . . . . . 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 6028 . . . . . . . . . 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 6070 . . . . . . . 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 2380 . . . . . . . 8  |-  (/)  =  (/)
35 seqomeq12 6640 . . . . . . . 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 2430 . . . . . 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 5005 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) ) )  ->  dom  h  =  dom  G )
4038, 39fveq12d 5667 . . . . 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 3229 . . . 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 2380 . . . 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 5675 . . . 4  |-  ( H `
 dom  G )  e.  _V
4441, 42, 43fvmpt 5738 . . 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 2412 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2646   _Vcvv 2892   [_csb 3187    \ cdif 3253   (/)c0 3564    e. cmpt 4200    _E cep 4426   Oncon0 4515   `'ccnv 4810   dom cdm 4811   "cima 4814   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015  seq𝜔cseqom 6633   1oc1o 6646    +o coa 6650    .o comu 6651    ^o coe 6652    ^m cmap 6947   Fincfn 7038  OrdIsocoi 7404   CNF ccnf 7542
This theorem is referenced by:  cantnfval2  7550  cantnfle  7552  cantnflt2  7554  cantnff  7555  cantnf0  7556  cantnfp1lem3  7562  cantnflem1  7571  cantnf  7575  cnfcom2  7585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-seqom 6634  df-oi 7405  df-cnf 7543
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