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Theorem cantnffval 8081
Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnffval.s  |-  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
cantnffval.a  |-  ( ph  ->  A  e.  On )
cantnffval.b  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnffval  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  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.a . 2  |-  ( ph  ->  A  e.  On )
2 cantnffval.b . 2  |-  ( ph  ->  B  e.  On )
3 oveq12 6294 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  ^m  y
)  =  ( A  ^m  B ) )
4 rabeq 3107 . . . . . 6  |-  ( ( x  ^m  y )  =  ( A  ^m  B )  ->  { g  e.  ( x  ^m  y )  |  g finSupp  (/)
}  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
53, 4syl 16 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  g finSupp  (/) }  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) } )
6 cantnffval.s . . . . 5  |-  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
75, 6syl6eqr 2526 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  g finSupp  (/) }  =  S )
8 simp1l 1020 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  x  =  A )
98oveq1d 6300 . . . . . . . . . 10  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( x  ^o  ( h `  k ) )  =  ( A  ^o  (
h `  k )
) )
109oveq1d 6300 . . . . . . . . 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 6300 . . . . . . . 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 6346 . . . . . . 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 2467 . . . . . . 7  |-  (/)  =  (/)
14 seqomeq12 7120 . . . . . . 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 662 . . . . . 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 5868 . . . . 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 3835 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  =  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) )
187, 17mpteq12dv 4525 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( f  e.  {
g  e.  ( x  ^m  y )  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  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 supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) )
19 df-cnf 8080 . . 3  |- CNF  =  ( x  e.  On , 
y  e.  On  |->  ( f  e.  { g  e.  ( x  ^m  y )  |  g finSupp  (/)
}  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( x  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) )
20 ovex 6310 . . . . . 6  |-  ( A  ^m  B )  e. 
_V
2120rabex 4598 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}  e.  _V
226, 21eqeltri 2551 . . . 4  |-  S  e. 
_V
2322mptex 6132 . . 3  |-  ( f  e.  S  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  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 6418 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) )
251, 2, 24syl2anc 661 1  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   [_csb 3435   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505    _E cep 4789   Oncon0 4878   dom cdm 4999   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   supp csupp 6902  seq𝜔cseqom 7113    +o coa 7128    .o comu 7129    ^o coe 7130    ^m cmap 7421   finSupp cfsupp 7830  OrdIsocoi 7935   CNF ccnf 8079
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-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-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-ov 6288  df-oprab 6289  df-mpt2 6290  df-recs 7043  df-rdg 7077  df-seqom 7114  df-cnf 8080
This theorem is referenced by:  cantnfdm  8082  cantnffvalOLD  8083  cantnfval  8088  cantnff  8094
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