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Theorem cantnff 8091
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 5862 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
21csbex 4566 . . 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 2441 . . . 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 8078 . . 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 8079 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
108, 9syl5eq 2494 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
1110mpteq1d 4514 . . 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 2485 . 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 2441 . . . . . . . 8  |- OrdIso (  _E  ,  ( x supp  (/) ) )  = OrdIso (  _E  , 
( x supp  (/) ) )
16 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
17 eqid 2441 . . . . . . . 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 8085 . . . . . . 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 6305 . . . . . . . . . . 11  |-  ( x supp  (/) )  e.  _V
218, 13, 14, 15, 16cantnfcl 8084 . . . . . . . . . . . 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 7961 . . . . . . . . . . 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 6912 . . . . . . . . . . . 12  |-  ( x supp  (/) )  C_  dom  x
278, 5, 6cantnfs 8083 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  x finSupp 
(/) ) ) )
2827simprbda 623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
29 fdm 5721 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
3126, 30syl5sseq 3534 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x supp  (/) )  C_  B
)
3231adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  C_  B
)
33 feq3 5701 . . . . . . . . . . . . . 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 5753 . . . . . . . . . . . 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 3799 . . . . . . . . . 10  |-  ( ( ( x supp  (/) )  C_  B  /\  B  =  (/) )  ->  ( x supp  (/) )  =  (/) )
4032, 38, 39syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  =  (/) )
4125, 40breqtrd 4457 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (/) )
42 en0 7576 . . . . . . . 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 5856 . . . . . 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 4563 . . . . . . 7  |-  (/)  e.  _V
4617seqom0g 7119 . . . . . . 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 2486 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
49 el1o 7147 . . . . 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 6293 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
5213adantr 465 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
53 oe0 7170 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
5452, 53syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
5551, 54eqtrd 2482 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
5650, 55eleqtrrd 2532 . . 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 4919 . . . . . 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 8090 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6556, 64pm2.61dane 2759 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
663, 12, 65fmpt2d 6042 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   {crab 2795   _Vcvv 3093   [_csb 3417    C_ wss 3458   (/)c0 3767   class class class wbr 4433    |-> cmpt 4491    _E cep 4775    We wwe 4823   Oncon0 4864   dom cdm 4985   -->wf 5570   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   omcom 6681   supp csupp 6899  seq𝜔cseqom 7110   1oc1o 7121    +o coa 7125    .o comu 7126    ^o coe 7127    ^m cmap 7418    ~~ cen 7511   finSupp cfsupp 7827  OrdIsocoi 7932   CNF ccnf 8076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-seqom 7111  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-oexp 7134  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-cnf 8077
This theorem is referenced by:  cantnfp1  8098  cantnflem1  8106  cantnflem3  8108  cantnflem4  8109  cantnf  8110  cantnfp1OLD  8124  cantnflem1OLD  8129  cantnflem3OLD  8130  cantnflem4OLD  8131  cantnfOLD  8132
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