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

Theorem cantnff 7870
Description: The CNF function is a function from finitely supported functions from  B to  A, to the ordinal exponential  A  ^o  B. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnff  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )

Proof of Theorem cantnff
Dummy variables  f 
g  h  k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5689 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
21csbex 4413 . . 3  |-  [_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
32a1i 11 . 2  |-  ( (
ph  /\  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 )
4 eqid 2433 . . . 4  |-  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}
5 cantnfs.a . . . 4  |-  ( ph  ->  A  e.  On )
6 cantnfs.b . . . 4  |-  ( ph  ->  B  e.  On )
74, 5, 6cantnffval 7857 . . 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
) ) )
8 cantnfs.s . . . . 5  |-  S  =  dom  ( A CNF  B
)
94, 5, 6cantnfdm 7858 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
108, 9syl5eq 2477 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
1110mpteq1d 4361 . . 3  |-  ( ph  ->  ( 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
) )  =  ( 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 ) ) )
127, 11eqtr4d 2468 . 2  |-  ( 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 ) ) )
135adantr 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  On )
146adantr 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  On )
15 eqid 2433 . . . . . . . 8  |- OrdIso (  _E  ,  ( x supp  (/) ) )  = OrdIso (  _E  , 
( x supp  (/) ) )
16 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
17 eqid 2433 . . . . . . . 8  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  (
x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )
188, 13, 14, 15, 16, 17cantnfval 7864 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( x supp  (/) ) ) ) )
1918adantr 462 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( x supp  (/) ) ) ) )
20 ovex 6105 . . . . . . . . . . 11  |-  ( x supp  (/) )  e.  _V
218, 13, 14, 15, 16cantnfcl 7863 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  (  _E  We  ( x supp  (/) )  /\  dom OrdIso (  _E  ,  ( x supp  (/) ) )  e. 
om ) )
2221simpld 456 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  _E  We  ( x supp  (/) ) )
2315oien 7740 . . . . . . . . . . 11  |-  ( ( ( x supp  (/) )  e. 
_V  /\  _E  We  ( x supp  (/) ) )  ->  dom OrdIso (  _E  , 
( x supp  (/) ) ) 
~~  ( x supp  (/) ) )
2420, 22, 23sylancr 656 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
2524adantr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
26 suppssdm 6692 . . . . . . . . . . . 12  |-  ( x supp  (/) )  C_  dom  x
278, 5, 6cantnfs 7862 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  x finSupp 
(/) ) ) )
2827simprbda 618 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
29 fdm 5551 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
3126, 30syl5sseq 3392 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x supp  (/) )  C_  B
)
3231adantr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  C_  B
)
33 feq3 5532 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  ->  ( x : B --> A  <->  x : B
--> (/) ) )
3428, 33syl5ibcom 220 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  ( A  =  (/)  ->  x : B --> (/) ) )
3534imp 429 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  x : B --> (/) )
36 f00 5581 . . . . . . . . . . . 12  |-  ( x : B --> (/)  <->  ( x  =  (/)  /\  B  =  (/) ) )
3735, 36sylib 196 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x  =  (/)  /\  B  =  (/) ) )
3837simprd 460 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  B  =  (/) )
39 sseq0 3657 . . . . . . . . . 10  |-  ( ( ( x supp  (/) )  C_  B  /\  B  =  (/) )  ->  ( x supp  (/) )  =  (/) )
4032, 38, 39syl2anc 654 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  =  (/) )
4125, 40breqtrd 4304 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (/) )
42 en0 7360 . . . . . . . 8  |-  ( dom OrdIso (  _E  ,  (
x supp  (/) ) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (
x supp  (/) ) )  =  (/) )
4341, 42sylib 196 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  =  (/) )
4443fveq2d 5683 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( x supp  (/) ) ) )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  (/) ) )
45 0ex 4410 . . . . . . 7  |-  (/)  e.  _V
4617seqom0g 6897 . . . . . . 7  |-  ( (/)  e.  _V  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  (/) )  =  (/) )
4745, 46mp1i 12 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( x supp  (/) ) ) `
 k ) )  .o  ( x `  (OrdIso (  _E  ,  ( x supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  (/) )  =  (/) )
4819, 44, 473eqtrd 2469 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
49 el1o 6927 . . . . 5  |-  ( ( ( A CNF  B ) `
 x )  e.  1o  <->  ( ( A CNF 
B ) `  x
)  =  (/) )
5048, 49sylibr 212 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  1o )
5138oveq2d 6096 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
5213adantr 462 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
53 oe0 6950 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
5452, 53syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
5551, 54eqtrd 2465 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
5650, 55eleqtrrd 2510 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
5713adantr 462 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  A  e.  On )
5814adantr 462 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  B  e.  On )
5916adantr 462 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  x  e.  S )
60 on0eln0 4761 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6113, 60syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6261biimpar 482 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (/)  e.  A
)
6331adantr 462 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
x supp  (/) )  C_  B
)
648, 57, 58, 59, 62, 58, 63cantnflt2 7869 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6556, 64pm2.61dane 2679 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
663, 12, 65fmpt2d 5860 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   {crab 2709   _Vcvv 2962   [_csb 3276    C_ wss 3316   (/)c0 3625   class class class wbr 4280    e. cmpt 4338    _E cep 4617    We wwe 4665   Oncon0 4706   dom cdm 4827   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   omcom 6465   supp csupp 6679  seq𝜔cseqom 6888   1oc1o 6901    +o coa 6905    .o comu 6906    ^o coe 6907    ^m cmap 7202    ~~ cen 7295   finSupp cfsupp 7608  OrdIsocoi 7711   CNF ccnf 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-seqom 6889  df-1o 6908  df-2o 6909  df-oadd 6912  df-omul 6913  df-oexp 6914  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-oi 7712  df-cnf 7856
This theorem is referenced by:  cantnfp1  7877  cantnflem1  7885  cantnflem3  7887  cantnflem4  7888  cantnf  7889  cantnfp1OLD  7903  cantnflem1OLD  7908  cantnflem3OLD  7909  cantnflem4OLD  7910  cantnfOLD  7911
  Copyright terms: Public domain W3C validator