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Theorem cantnffval 7364
Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnffval.1  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
cantnffval.2  |-  ( ph  ->  A  e.  On )
cantnffval.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnffval  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_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
) ) )
Distinct variable groups:    f, g, h, k, z, A    B, f, g, h, k, z    S, f
Allowed substitution hints:    ph( z, f, g, h, k)    S( z, g, h, k)

Proof of Theorem cantnffval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.2 . 2  |-  ( ph  ->  A  e.  On )
2 cantnffval.3 . 2  |-  ( ph  ->  B  e.  On )
3 oveq12 5867 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  ^m  y
)  =  ( A  ^m  B ) )
4 rabeq 2782 . . . . . 6  |-  ( ( x  ^m  y )  =  ( A  ^m  B )  ->  { g  e.  ( x  ^m  y )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin } )
53, 4syl 15 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } )
6 cantnffval.1 . . . . 5  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
75, 6syl6eqr 2333 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }  =  S )
8 simp1l 979 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  x  =  A )
98oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( x  ^o  ( h `  k ) )  =  ( A  ^o  (
h `  k )
) )
109oveq1d 5873 . . . . . . . . 9  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( ( x  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  =  ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) ) )
1110oveq1d 5873 . . . . . . . 8  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( ( ( x  ^o  (
h `  k )
)  .o  ( f `
 ( h `  k ) ) )  +o  z )  =  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) )
1211mpt2eq3dva 5912 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) )  =  ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) )
13 eqid 2283 . . . . . . 7  |-  (/)  =  (/)
14 seqomeq12 6466 . . . . . . 7  |-  ( ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) )  =  ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) )  /\  (/)  =  (/) )  -> seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) )
1512, 13, 14sylancl 643 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  -> seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) )
1615fveq1d 5527 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( x  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  =  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h ) )
1716csbeq2dv 3106 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( x  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  =  [_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
) )
187, 17mpteq12dv 4098 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( f  e.  {
g  e.  ( x  ^m  y )  |  ( `' g "
( _V  \  1o ) )  e.  Fin } 
|->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( x  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  ( f  e.  S  |->  [_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
) ) )
19 df-cnf 7363 . . 3  |- CNF  =  ( x  e.  On , 
y  e.  On  |->  ( f  e.  { g  e.  ( x  ^m  y )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( x  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
20 ovex 5883 . . . . . 6  |-  ( A  ^m  B )  e. 
_V
2120rabex 4165 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  e.  _V
226, 21eqeltri 2353 . . . 4  |-  S  e. 
_V
2322mptex 5746 . . 3  |-  ( f  e.  S  |->  [_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
) )  e.  _V
2418, 19, 23ovmpt2a 5978 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_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
) ) )
251, 2, 24syl2anc 642 1  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_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
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    \ cdif 3149   (/)c0 3455    e. cmpt 4077    _E cep 4303   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478    ^m cmap 6772   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnfdm  7365  cantnfval  7369  cantnff  7375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seqom 6460  df-cnf 7363
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