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Theorem cantnfval 8076
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
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfcl.g  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
cantnfcl.f  |-  ( ph  ->  F  e.  S )
cantnfval.h  |-  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 2460 . . . 4  |-  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}
2 cantnfs.a . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfs.b . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffval 8069 . . 3  |-  ( ph  ->  ( A CNF  B )  =  ( f  e. 
{ g  e.  ( A  ^m  B )  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
54fveq1d 5859 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( ( f  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) `  F
) )
6 cantnfcl.f . . . 4  |-  ( ph  ->  F  e.  S )
7 cantnfs.s . . . . 5  |-  S  =  dom  ( A CNF  B
)
81, 2, 3cantnfdm 8070 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
97, 8syl5eq 2513 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
106, 9eleqtrd 2550 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
11 ovex 6300 . . . . . 6  |-  ( f supp  (/) )  e.  _V
12 eqid 2460 . . . . . . 7  |- OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso (  _E  , 
( f supp  (/) ) )
1312oiexg 7949 . . . . . 6  |-  ( ( f supp  (/) )  e.  _V  -> OrdIso (  _E  ,  ( f supp  (/) ) )  e. 
_V )
1411, 13mp1i 12 . . . . 5  |-  ( f  =  F  -> OrdIso (  _E  ,  ( f supp  (/) ) )  e.  _V )
15 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  = OrdIso (  _E  ,  ( f supp  (/) ) ) )
16 oveq1 6282 . . . . . . . . . . . . . . . . 17  |-  ( f  =  F  ->  (
f supp  (/) )  =  ( F supp  (/) ) )
1716adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f supp  (/) )  =  ( F supp  (/) ) )
18 oieq2 7927 . . . . . . . . . . . . . . . 16  |-  ( ( f supp  (/) )  =  ( F supp  (/) )  -> OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) ) )
1917, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  -> OrdIso (  _E  , 
( f supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) ) )
2015, 19eqtrd 2501 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  = OrdIso (  _E  ,  ( F supp  (/) ) ) )
21 cantnfcl.g . . . . . . . . . . . . . 14  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
2220, 21syl6eqr 2519 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  =  G )
2322fveq1d 5859 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2423oveq2d 6291 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
25 simpl 457 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  f  =  F )
2625, 23fveq12d 5863 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
2724, 26oveq12d 6293 . . . . . . . . . 10  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  =  ( ( A  ^o  ( G `  k )
)  .o  ( F `
 ( G `  k ) ) ) )
2827oveq1d 6290 . . . . . . . . 9  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( (
( A  ^o  (
h `  k )
)  .o  ( f `
 ( h `  k ) ) )  +o  z )  =  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) )
2928mpt2eq3dv 6338 . . . . . . . 8  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( 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
) ) )
30 eqid 2460 . . . . . . . 8  |-  (/)  =  (/)
31 seqomeq12 7109 . . . . . . . 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
) ) ,  (/) ) )
3229, 30, 31sylancl 662 . . . . . . 7  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  -> 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
) ) ,  (/) ) )
33 cantnfval.h . . . . . . 7  |-  H  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( G `  k ) )  .o  ( F `  ( G `  k )
) )  +o  z
) ) ,  (/) )
3432, 33syl6eqr 2519 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  -> seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) )  =  H
)
3522dmeqd 5196 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  dom  h  =  dom  G )
3634, 35fveq12d 5863 . . . . 5  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  =  ( H `
 dom  G )
)
3714, 36csbied 3455 . . . 4  |-  ( f  =  F  ->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  =  ( H `  dom  G ) )
38 eqid 2460 . . . 4  |-  ( f  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  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 finSupp  (/)
}  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) )
39 fvex 5867 . . . 4  |-  ( H `
 dom  G )  e.  _V
4037, 38, 39fvmpt 5941 . . 3  |-  ( F  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) }  ->  ( ( f  e.  {
g  e.  ( A  ^m  B )  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) `  F )  =  ( H `  dom  G ) )
4110, 40syl 16 . 2  |-  ( ph  ->  ( ( f  e. 
{ g  e.  ( A  ^m  B )  |  g finSupp  (/) }  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) `  F
)  =  ( H `
 dom  G )
)
425, 41eqtrd 2501 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106   [_csb 3428   (/)c0 3778   class class class wbr 4440    |-> cmpt 4498    _E cep 4782   Oncon0 4871   dom cdm 4992   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   supp csupp 6891  seq𝜔cseqom 7102    +o coa 7117    .o comu 7118    ^o coe 7119    ^m cmap 7410   finSupp cfsupp 7818  OrdIsocoi 7923   CNF ccnf 8067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-recs 7032  df-rdg 7066  df-seqom 7103  df-oi 7924  df-cnf 8068
This theorem is referenced by:  cantnfval2  8077  cantnfle  8079  cantnflt2  8081  cantnff  8082  cantnf0  8083  cantnfp1lem3  8088  cantnflem1  8097  cantnf  8101  cnfcom2  8135
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