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Theorem cantnff 8110
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 5882 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
21csbex 4590 . . 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 2457 . . . 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 8097 . . 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 8098 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
108, 9syl5eq 2510 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
1110mpteq1d 4538 . . 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 2501 . 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 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  On )
146adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  On )
15 eqid 2457 . . . . . . . 8  |- OrdIso (  _E  ,  ( x supp  (/) ) )  = OrdIso (  _E  , 
( x supp  (/) ) )
16 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
17 eqid 2457 . . . . . . . 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 8104 . . . . . . 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 465 . . . . . 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 6324 . . . . . . . . . . 11  |-  ( x supp  (/) )  e.  _V
218, 13, 14, 15, 16cantnfcl 8103 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  (  _E  We  ( x supp  (/) )  /\  dom OrdIso (  _E  ,  ( x supp  (/) ) )  e. 
om ) )
2221simpld 459 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  _E  We  ( x supp  (/) ) )
2315oien 7981 . . . . . . . . . . 11  |-  ( ( ( x supp  (/) )  e. 
_V  /\  _E  We  ( x supp  (/) ) )  ->  dom OrdIso (  _E  , 
( x supp  (/) ) ) 
~~  ( x supp  (/) ) )
2420, 22, 23sylancr 663 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
2524adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
26 suppssdm 6930 . . . . . . . . . . . 12  |-  ( x supp  (/) )  C_  dom  x
278, 5, 6cantnfs 8102 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  x finSupp 
(/) ) ) )
2827simprbda 623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
29 fdm 5741 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
3126, 30syl5sseq 3547 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x supp  (/) )  C_  B
)
3231adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  C_  B
)
33 feq3 5721 . . . . . . . . . . . . . 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 5773 . . . . . . . . . . . 12  |-  ( x : B --> (/)  <->  ( x  =  (/)  /\  B  =  (/) ) )
3735, 36sylib 196 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x  =  (/)  /\  B  =  (/) ) )
3837simprd 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  B  =  (/) )
39 sseq0 3826 . . . . . . . . . 10  |-  ( ( ( x supp  (/) )  C_  B  /\  B  =  (/) )  ->  ( x supp  (/) )  =  (/) )
4032, 38, 39syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  =  (/) )
4125, 40breqtrd 4480 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (/) )
42 en0 7597 . . . . . . . 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 5876 . . . . . 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 4587 . . . . . . 7  |-  (/)  e.  _V
4617seqom0g 7139 . . . . . . 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 2502 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
49 el1o 7167 . . . . 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 6312 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
5213adantr 465 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
53 oe0 7190 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
5452, 53syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
5551, 54eqtrd 2498 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
5650, 55eleqtrrd 2548 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
5713adantr 465 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  A  e.  On )
5814adantr 465 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  B  e.  On )
5916adantr 465 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  x  e.  S )
60 on0eln0 4942 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6113, 60syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6261biimpar 485 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (/)  e.  A
)
6331adantr 465 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
x supp  (/) )  C_  B
)
648, 57, 58, 59, 62, 58, 63cantnflt2 8109 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6556, 64pm2.61dane 2775 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
663, 12, 65fmpt2d 6062 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   [_csb 3430    C_ wss 3471   (/)c0 3793   class class class wbr 4456    |-> cmpt 4515    _E cep 4798    We wwe 4846   Oncon0 4887   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   omcom 6699   supp csupp 6917  seq𝜔cseqom 7130   1oc1o 7141    +o coa 7145    .o comu 7146    ^o coe 7147    ^m cmap 7438    ~~ cen 7532   finSupp cfsupp 7847  OrdIsocoi 7952   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnfp1  8117  cantnflem1  8125  cantnflem3  8127  cantnflem4  8128  cantnf  8129  cantnfp1OLD  8143  cantnflem1OLD  8148  cantnflem3OLD  8149  cantnflem4OLD  8150  cantnfOLD  8151
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