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Theorem cantnf0 8097
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 2443 . . 3  |- OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )  = OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )
5 cantnf0.a . . . . 5  |-  ( ph  -> 
(/)  e.  A )
6 fconst6g 5764 . . . . 5  |-  ( (/)  e.  A  ->  ( B  X.  { (/) } ) : B --> A )
75, 6syl 16 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> A )
83, 5fczfsuppd 7849 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) finSupp  (/) )
91, 2, 3cantnfs 8088 . . . 4  |-  ( ph  ->  ( ( B  X.  { (/) } )  e.  S  <->  ( ( B  X.  { (/) } ) : B --> A  /\  ( B  X.  { (/) } ) finSupp  (/) ) ) )
107, 8, 9mpbir2and 922 . . 3  |-  ( ph  ->  ( B  X.  { (/)
} )  e.  S
)
11 eqid 2443 . . 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 8090 . 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 2444 . . . . . . 7  |-  ( ph  ->  ( B  X.  { (/)
} )  =  ( B  X.  { (/) } ) )
14 0ex 4567 . . . . . . . . 9  |-  (/)  e.  _V
15 fnconstg 5763 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( B  X.  { (/) } )  Fn  B )
1614, 15mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( B  X.  { (/)
} )  Fn  B
)
1714a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
18 fnsuppeq0 6930 . . . . . . . 8  |-  ( ( ( B  X.  { (/)
} )  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  -> 
( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
1916, 3, 17, 18syl3anc 1229 . . . . . . 7  |-  ( ph  ->  ( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
2013, 19mpbird 232 . . . . . 6  |-  ( ph  ->  ( ( B  X.  { (/) } ) supp  (/) )  =  (/) )
21 oieq2 7941 . . . . . 6  |-  ( ( ( B  X.  { (/)
} ) supp  (/) )  =  (/)  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2220, 21syl 16 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2322dmeqd 5195 . . . 4  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  dom OrdIso (  _E  ,  (/) ) )
24 we0 4864 . . . . . 6  |-  _E  We  (/)
25 eqid 2443 . . . . . . 7  |- OrdIso (  _E  ,  (/) )  = OrdIso (  _E  ,  (/) )
2625oien 7966 . . . . . 6  |-  ( (
(/)  e.  _V  /\  _E  We  (/) )  ->  dom OrdIso (  _E  ,  (/) )  ~~  (/) )
2714, 24, 26mp2an 672 . . . . 5  |-  dom OrdIso (  _E  ,  (/) )  ~~  (/)
28 en0 7580 . . . . 5  |-  ( dom OrdIso (  _E  ,  (/) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (/) )  =  (/) )
2927, 28mpbi 208 . . . 4  |-  dom OrdIso (  _E  ,  (/) )  =  (/)
3023, 29syl6eq 2500 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  (/) )
3130fveq2d 5860 . 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 7123 . . 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 2488 1  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804   _Vcvv 3095   (/)c0 3770   {csn 4014   class class class wbr 4437    _E cep 4779    We wwe 4827   Oncon0 4868    X. cxp 4987   dom cdm 4989    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   supp csupp 6903  seq𝜔cseqom 7114    +o coa 7129    .o comu 7130    ^o coe 7131    ~~ cen 7515   finSupp cfsupp 7831  OrdIsocoi 7937   CNF ccnf 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-map 7424  df-en 7519  df-fin 7522  df-fsupp 7832  df-oi 7938  df-cnf 8082
This theorem is referenced by:  cnfcom2lem  8148  cnfcom2lemOLD  8156
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