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Theorem cantnff 8184
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 5880 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
21csbex 4541 . . 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 2453 . . . 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 8173 . . 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 8174 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
108, 9syl5eq 2499 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
1110mpteq1d 4487 . . 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 2490 . 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 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  On )
146adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  On )
15 eqid 2453 . . . . . . . 8  |- OrdIso (  _E  ,  ( x supp  (/) ) )  = OrdIso (  _E  , 
( x supp  (/) ) )
16 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
17 eqid 2453 . . . . . . . 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 8178 . . . . . . 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 467 . . . . . 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 6323 . . . . . . . . . . 11  |-  ( x supp  (/) )  e.  _V
218, 13, 14, 15, 16cantnfcl 8177 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  (  _E  We  ( x supp  (/) )  /\  dom OrdIso (  _E  ,  ( x supp  (/) ) )  e. 
om ) )
2221simpld 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  _E  We  ( x supp  (/) ) )
2315oien 8058 . . . . . . . . . . 11  |-  ( ( ( x supp  (/) )  e. 
_V  /\  _E  We  ( x supp  (/) ) )  ->  dom OrdIso (  _E  , 
( x supp  (/) ) ) 
~~  ( x supp  (/) ) )
2420, 22, 23sylancr 670 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
2524adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (
x supp  (/) ) )
26 suppssdm 6932 . . . . . . . . . . . 12  |-  ( x supp  (/) )  C_  dom  x
278, 5, 6cantnfs 8176 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  x finSupp 
(/) ) ) )
2827simprbda 629 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
29 fdm 5738 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3028, 29syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
3126, 30syl5sseq 3482 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x supp  (/) )  C_  B
)
3231adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  C_  B
)
33 feq3 5717 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  ->  ( x : B --> A  <->  x : B
--> (/) ) )
3428, 33syl5ibcom 224 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  ( A  =  (/)  ->  x : B --> (/) ) )
3534imp 431 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  x : B --> (/) )
36 f00 5770 . . . . . . . . . . . 12  |-  ( x : B --> (/)  <->  ( x  =  (/)  /\  B  =  (/) ) )
3735, 36sylib 200 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x  =  (/)  /\  B  =  (/) ) )
3837simprd 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  B  =  (/) )
39 sseq0 3768 . . . . . . . . . 10  |-  ( ( ( x supp  (/) )  C_  B  /\  B  =  (/) )  ->  ( x supp  (/) )  =  (/) )
4032, 38, 39syl2anc 667 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x supp  (/) )  =  (/) )
4125, 40breqtrd 4430 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  ~~  (/) )
42 en0 7637 . . . . . . . 8  |-  ( dom OrdIso (  _E  ,  (
x supp  (/) ) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (
x supp  (/) ) )  =  (/) )
4341, 42sylib 200 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( x supp  (/) ) )  =  (/) )
4443fveq2d 5874 . . . . . 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 4538 . . . . . . 7  |-  (/)  e.  _V
4617seqom0g 7178 . . . . . . 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 13 . . . . . 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 2491 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
49 el1o 7206 . . . . 5  |-  ( ( ( A CNF  B ) `
 x )  e.  1o  <->  ( ( A CNF 
B ) `  x
)  =  (/) )
5048, 49sylibr 216 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  1o )
5138oveq2d 6311 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
5213adantr 467 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
53 oe0 7229 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
5452, 53syl 17 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
5551, 54eqtrd 2487 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
5650, 55eleqtrrd 2534 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
5713adantr 467 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  A  e.  On )
5814adantr 467 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  B  e.  On )
5916adantr 467 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  x  e.  S )
60 on0eln0 5481 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6113, 60syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
6261biimpar 488 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (/)  e.  A
)
6331adantr 467 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
x supp  (/) )  C_  B
)
648, 57, 58, 59, 62, 58, 63cantnflt2 8183 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6556, 64pm2.61dane 2713 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
663, 12, 65fmpt2d 6058 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889    =/= wne 2624   {crab 2743   _Vcvv 3047   [_csb 3365    C_ wss 3406   (/)c0 3733   class class class wbr 4405    |-> cmpt 4464    _E cep 4746    We wwe 4795   dom cdm 4837   Oncon0 5426   -->wf 5581   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   omcom 6697   supp csupp 6919  seq𝜔cseqom 7169   1oc1o 7180    +o coa 7184    .o comu 7185    ^o coe 7186    ^m cmap 7477    ~~ cen 7571   finSupp cfsupp 7888  OrdIsocoi 8029   CNF ccnf 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-seqom 7170  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-oexp 7193  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-oi 8030  df-cnf 8172
This theorem is referenced by:  cantnfp1  8191  cantnflem1  8199  cantnflem3  8201  cantnflem4  8202  cantnf  8203
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