MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfval Structured version   Unicode version

Theorem cantnfval 7876
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 2443 . . . 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 7869 . . 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 5693 . 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 7870 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
97, 8syl5eq 2487 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
106, 9eleqtrd 2519 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
11 ovex 6116 . . . . . 6  |-  ( f supp  (/) )  e.  _V
12 eqid 2443 . . . . . . 7  |- OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso (  _E  , 
( f supp  (/) ) )
1312oiexg 7749 . . . . . 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 6098 . . . . . . . . . . . . . . . . 17  |-  ( f  =  F  ->  (
f supp  (/) )  =  ( F supp  (/) ) )
1716adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f supp  (/) )  =  ( F supp  (/) ) )
18 oieq2 7727 . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  = OrdIso (  _E  ,  ( F supp  (/) ) ) )
21 cantnfcl.g . . . . . . . . . . . . . 14  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
2220, 21syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  =  G )
2322fveq1d 5693 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2423oveq2d 6107 . . . . . . . . . . 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 5697 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
2724, 26oveq12d 6109 . . . . . . . . . 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 6106 . . . . . . . . 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 6152 . . . . . . . 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 2443 . . . . . . . 8  |-  (/)  =  (/)
31 seqomeq12 6909 . . . . . . . 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 2493 . . . . . 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 5042 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  dom  h  =  dom  G )
3634, 35fveq12d 5697 . . . . 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 3314 . . . 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 2443 . . . 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 5701 . . . 4  |-  ( H `
 dom  G )  e.  _V
4037, 38, 39fvmpt 5774 . . 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 2475 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972   [_csb 3288   (/)c0 3637   class class class wbr 4292    e. cmpt 4350    _E cep 4630   Oncon0 4719   dom cdm 4840   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   supp csupp 6690  seq𝜔cseqom 6902    +o coa 6917    .o comu 6918    ^o coe 6919    ^m cmap 7214   finSupp cfsupp 7620  OrdIsocoi 7723   CNF ccnf 7867
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-recs 6832  df-rdg 6866  df-seqom 6903  df-oi 7724  df-cnf 7868
This theorem is referenced by:  cantnfval2  7877  cantnfle  7879  cantnflt2  7881  cantnff  7882  cantnf0  7883  cantnfp1lem3  7888  cantnflem1  7897  cantnf  7901  cnfcom2  7935
  Copyright terms: Public domain W3C validator