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Theorem cantnff 7997
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 5812 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
21csbex 4536 . . 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 2454 . . . 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 7984 . . 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 7985 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
108, 9syl5eq 2507 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
1110mpteq1d 4484 . . 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 2498 . 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 2454 . . . . . . . 8  |- OrdIso (  _E  ,  ( x supp  (/) ) )  = OrdIso (  _E  , 
( x supp  (/) ) )
16 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
17 eqid 2454 . . . . . . . 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 7991 . . . . . . 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 6228 . . . . . . . . . . 11  |-  ( x supp  (/) )  e.  _V
218, 13, 14, 15, 16cantnfcl 7990 . . . . . . . . . . . 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 7867 . . . . . . . . . . 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 6816 . . . . . . . . . . . 12  |-  ( x supp  (/) )  C_  dom  x
278, 5, 6cantnfs 7989 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  x finSupp 
(/) ) ) )
2827simprbda 623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
29 fdm 5674 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
3126, 30syl5sseq 3515 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x supp  (/) )  C_  B
)
3231adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  C_  B
)
33 feq3 5655 . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . 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 3780 . . . . . . . . . 10  |-  ( ( ( x supp  (/) )  C_  B  /\  B  =  (/) )  ->  ( x supp  (/) )  =  (/) )
4032, 38, 39syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  =  (/) )
4125, 40breqtrd 4427 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (/) )
42 en0 7485 . . . . . . . 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 5806 . . . . . 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 4533 . . . . . . 7  |-  (/)  e.  _V
4617seqom0g 7024 . . . . . . 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 2499 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
49 el1o 7052 . . . . 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 6219 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
5213adantr 465 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
53 oe0 7075 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
5452, 53syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
5551, 54eqtrd 2495 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
5650, 55eleqtrrd 2545 . . 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 4885 . . . . . 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 7996 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6556, 64pm2.61dane 2770 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
663, 12, 65fmpt2d 5985 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078   [_csb 3398    C_ wss 3439   (/)c0 3748   class class class wbr 4403    |-> cmpt 4461    _E cep 4741    We wwe 4789   Oncon0 4830   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   omcom 6589   supp csupp 6803  seq𝜔cseqom 7015   1oc1o 7026    +o coa 7030    .o comu 7031    ^o coe 7032    ^m cmap 7327    ~~ cen 7420   finSupp cfsupp 7734  OrdIsocoi 7838   CNF ccnf 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-cnf 7983
This theorem is referenced by:  cantnfp1  8004  cantnflem1  8012  cantnflem3  8014  cantnflem4  8015  cantnf  8016  cantnfp1OLD  8030  cantnflem1OLD  8035  cantnflem3OLD  8036  cantnflem4OLD  8037  cantnfOLD  8038
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