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Theorem cantnf0 8106
Description: The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnf0.a  |-  ( ph  -> 
(/)  e.  A )
Assertion
Ref Expression
cantnf0  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )

Proof of Theorem cantnf0
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfs.s . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.a . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfs.b . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2467 . . 3  |- OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )  = OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )
5 cantnf0.a . . . . 5  |-  ( ph  -> 
(/)  e.  A )
6 fconst6g 5780 . . . . 5  |-  ( (/)  e.  A  ->  ( B  X.  { (/) } ) : B --> A )
75, 6syl 16 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> A )
83, 5fczfsuppd 7859 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) finSupp  (/) )
91, 2, 3cantnfs 8097 . . . 4  |-  ( ph  ->  ( ( B  X.  { (/) } )  e.  S  <->  ( ( B  X.  { (/) } ) : B --> A  /\  ( B  X.  { (/) } ) finSupp  (/) ) ) )
107, 8, 9mpbir2and 920 . . 3  |-  ( ph  ->  ( B  X.  { (/)
} )  e.  S
)
11 eqid 2467 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) )
121, 2, 3, 4, 10, 11cantnfval 8099 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) ) ) )
13 eqidd 2468 . . . . . . 7  |-  ( ph  ->  ( B  X.  { (/)
} )  =  ( B  X.  { (/) } ) )
14 0ex 4583 . . . . . . . . 9  |-  (/)  e.  _V
15 fnconstg 5779 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( B  X.  { (/) } )  Fn  B )
1614, 15mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( B  X.  { (/)
} )  Fn  B
)
1714a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
18 fnsuppeq0 6940 . . . . . . . 8  |-  ( ( ( B  X.  { (/)
} )  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  -> 
( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
1916, 3, 17, 18syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
2013, 19mpbird 232 . . . . . 6  |-  ( ph  ->  ( ( B  X.  { (/) } ) supp  (/) )  =  (/) )
21 oieq2 7950 . . . . . 6  |-  ( ( ( B  X.  { (/)
} ) supp  (/) )  =  (/)  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2220, 21syl 16 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2322dmeqd 5211 . . . 4  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  dom OrdIso (  _E  ,  (/) ) )
24 we0 4880 . . . . . 6  |-  _E  We  (/)
25 eqid 2467 . . . . . . 7  |- OrdIso (  _E  ,  (/) )  = OrdIso (  _E  ,  (/) )
2625oien 7975 . . . . . 6  |-  ( (
(/)  e.  _V  /\  _E  We  (/) )  ->  dom OrdIso (  _E  ,  (/) )  ~~  (/) )
2714, 24, 26mp2an 672 . . . . 5  |-  dom OrdIso (  _E  ,  (/) )  ~~  (/)
28 en0 7590 . . . . 5  |-  ( dom OrdIso (  _E  ,  (/) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (/) )  =  (/) )
2927, 28mpbi 208 . . . 4  |-  dom OrdIso (  _E  ,  (/) )  =  (/)
3023, 29syl6eq 2524 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  (/) )
3130fveq2d 5876 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) ) )  =  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) )  .o  ( ( B  X.  { (/) } ) `
 (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) ) )  +o  z ) ) ,  (/) ) `  (/) ) )
3211seqom0g 7133 . . 3  |-  ( (/)  e.  _V  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) ) `  (/) )  =  (/) )
3314, 32mp1i 12 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) ) `  k
) )  .o  (
( B  X.  { (/)
} ) `  (OrdIso (  _E  ,  (
( B  X.  { (/)
} ) supp  (/) ) ) `
 k ) ) )  +o  z ) ) ,  (/) ) `  (/) )  =  (/) )
3412, 31, 333eqtrd 2512 1  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4033   class class class wbr 4453    _E cep 4795    We wwe 4843   Oncon0 4884    X. cxp 5003   dom cdm 5005    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   supp csupp 6913  seq𝜔cseqom 7124    +o coa 7139    .o comu 7140    ^o coe 7141    ~~ cen 7525   finSupp cfsupp 7841  OrdIsocoi 7946   CNF ccnf 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-map 7434  df-en 7529  df-fin 7532  df-fsupp 7842  df-oi 7947  df-cnf 8091
This theorem is referenced by:  cnfcom2lem  8157  cnfcom2lemOLD  8165
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