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Theorem cantnfval 8118
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 2402 . . . 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 8111 . . 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 5850 . 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 8112 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
97, 8syl5eq 2455 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
106, 9eleqtrd 2492 . . 3  |-  ( ph  ->  F  e.  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
11 ovex 6305 . . . . . 6  |-  ( f supp  (/) )  e.  _V
12 eqid 2402 . . . . . . 7  |- OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso (  _E  , 
( f supp  (/) ) )
1312oiexg 7993 . . . . . 6  |-  ( ( f supp  (/) )  e.  _V  -> OrdIso (  _E  ,  ( f supp  (/) ) )  e. 
_V )
1411, 13mp1i 13 . . . . 5  |-  ( f  =  F  -> OrdIso (  _E  ,  ( f supp  (/) ) )  e.  _V )
15 simpr 459 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  = OrdIso (  _E  ,  ( f supp  (/) ) ) )
16 oveq1 6284 . . . . . . . . . . . . . . . . 17  |-  ( f  =  F  ->  (
f supp  (/) )  =  ( F supp  (/) ) )
1716adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f supp  (/) )  =  ( F supp  (/) ) )
18 oieq2 7971 . . . . . . . . . . . . . . . 16  |-  ( ( f supp  (/) )  =  ( F supp  (/) )  -> OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) ) )
1917, 18syl 17 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  -> OrdIso (  _E  , 
( f supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) ) )
2015, 19eqtrd 2443 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  = OrdIso (  _E  ,  ( F supp  (/) ) ) )
21 cantnfcl.g . . . . . . . . . . . . . 14  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
2220, 21syl6eqr 2461 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  h  =  G )
2322fveq1d 5850 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( h `  k )  =  ( G `  k ) )
2423oveq2d 6293 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( A  ^o  ( h `  k
) )  =  ( A  ^o  ( G `
 k ) ) )
25 simpl 455 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  f  =  F )
2625, 23fveq12d 5854 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  ( f `  ( h `  k
) )  =  ( F `  ( G `
 k ) ) )
2724, 26oveq12d 6295 . . . . . . . . . 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 6292 . . . . . . . . 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 6343 . . . . . . . 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 2402 . . . . . . . 8  |-  (/)  =  (/)
31 seqomeq12 7155 . . . . . . . 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 660 . . . . . . 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 2461 . . . . . 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 5025 . . . . . 6  |-  ( ( f  =  F  /\  h  = OrdIso (  _E  , 
( f supp  (/) ) ) )  ->  dom  h  =  dom  G )
3634, 35fveq12d 5854 . . . . 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 3399 . . . 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 2402 . . . 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 5858 . . . 4  |-  ( H `
 dom  G )  e.  _V
4037, 38, 39fvmpt 5931 . . 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 17 . 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 2443 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  ( H `  dom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2757   _Vcvv 3058   [_csb 3372   (/)c0 3737   class class class wbr 4394    |-> cmpt 4452    _E cep 4731   dom cdm 4822   Oncon0 5409   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   supp csupp 6901  seq𝜔cseqom 7148    +o coa 7163    .o comu 7164    ^o coe 7165    ^m cmap 7456   finSupp cfsupp 7862  OrdIsocoi 7967   CNF ccnf 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-oi 7968  df-cnf 8110
This theorem is referenced by:  cantnfval2  8119  cantnfle  8121  cantnflt2  8123  cantnff  8124  cantnf0  8125  cantnfp1lem3  8130  cantnflem1  8139  cantnf  8143  cnfcom2  8177
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