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Theorem cantnffval 7881
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 6112 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  ^m  y
)  =  ( A  ^m  B ) )
4 rabeq 2978 . . . . . 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 2493 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  g finSupp  (/) }  =  S )
8 simp1l 1012 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  x  =  A )
98oveq1d 6118 . . . . . . . . . 10  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( x  ^o  ( h `  k ) )  =  ( A  ^o  (
h `  k )
) )
109oveq1d 6118 . . . . . . . . 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 6118 . . . . . . . 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 6162 . . . . . . 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 2443 . . . . . . 7  |-  (/)  =  (/)
14 seqomeq12 6921 . . . . . . 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 5705 . . . . 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 3699 . . . 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 4382 . . 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 7880 . . 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 6128 . . . . . 6  |-  ( A  ^m  B )  e. 
_V
2120rabex 4455 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}  e.  _V
226, 21eqeltri 2513 . . . 4  |-  S  e. 
_V
2322mptex 5960 . . 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 6233 . 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 965    = wceq 1369    e. wcel 1756   {crab 2731   _Vcvv 2984   [_csb 3300   (/)c0 3649   class class class wbr 4304    e. cmpt 4362    _E cep 4642   Oncon0 4731   dom cdm 4852   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   supp csupp 6702  seq𝜔cseqom 6914    +o coa 6929    .o comu 6930    ^o coe 6931    ^m cmap 7226   finSupp cfsupp 7632  OrdIsocoi 7735   CNF ccnf 7879
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-recs 6844  df-rdg 6878  df-seqom 6915  df-cnf 7880
This theorem is referenced by:  cantnfdm  7882  cantnffvalOLD  7883  cantnfval  7888  cantnff  7894
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