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

Theorem cantnffval 8071
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 6279 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  ^m  y
)  =  ( A  ^m  B ) )
4 rabeq 3100 . . . . . 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 2513 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  { g  e.  ( x  ^m  y )  |  g finSupp  (/) }  =  S )
8 simp1l 1018 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  x  =  A )
98oveq1d 6285 . . . . . . . . . 10  |-  ( ( ( x  =  A  /\  y  =  B )  /\  k  e. 
_V  /\  z  e.  _V )  ->  ( x  ^o  ( h `  k ) )  =  ( A  ^o  (
h `  k )
) )
109oveq1d 6285 . . . . . . . . 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 6285 . . . . . . . 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 6334 . . . . . . 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 2454 . . . . . . 7  |-  (/)  =  (/)
14 seqomeq12 7111 . . . . . . 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 660 . . . . . 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 5850 . . . . 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 3831 . . . 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 4517 . . 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 8070 . . 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 6298 . . . . 5  |-  ( A  ^m  B )  e. 
_V
216, 20rabex2 4590 . . . 4  |-  S  e. 
_V
2221mptex 6118 . . 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
2318, 19, 22ovmpt2a 6406 . 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 ) ) )
241, 2, 23syl2anc 659 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106   [_csb 3420   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497    _E cep 4778   Oncon0 4867   dom cdm 4988   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   supp csupp 6891  seq𝜔cseqom 7104    +o coa 7119    .o comu 7120    ^o coe 7121    ^m cmap 7412   finSupp cfsupp 7821  OrdIsocoi 7926   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seqom 7105  df-cnf 8070
This theorem is referenced by:  cantnfdm  8072  cantnffvalOLD  8073  cantnfval  8078  cantnff  8084
  Copyright terms: Public domain W3C validator