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Theorem cantnf0 7997
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 2454 . . 3  |- OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )  = OrdIso (  _E  ,  ( ( B  X.  { (/) } ) supp  (/) ) )
5 cantnf0.a . . . . 5  |-  ( ph  -> 
(/)  e.  A )
6 fconst6g 5710 . . . . 5  |-  ( (/)  e.  A  ->  ( B  X.  { (/) } ) : B --> A )
75, 6syl 16 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> A )
83, 5fczfsuppd 7752 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) finSupp  (/) )
91, 2, 3cantnfs 7988 . . . 4  |-  ( ph  ->  ( ( B  X.  { (/) } )  e.  S  <->  ( ( B  X.  { (/) } ) : B --> A  /\  ( B  X.  { (/) } ) finSupp  (/) ) ) )
107, 8, 9mpbir2and 913 . . 3  |-  ( ph  ->  ( B  X.  { (/)
} )  e.  S
)
11 eqid 2454 . . 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 7990 . 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 2455 . . . . . . 7  |-  ( ph  ->  ( B  X.  { (/)
} )  =  ( B  X.  { (/) } ) )
14 0ex 4533 . . . . . . . . 9  |-  (/)  e.  _V
15 fnconstg 5709 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( B  X.  { (/) } )  Fn  B )
1614, 15mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( B  X.  { (/)
} )  Fn  B
)
1714a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
18 fnsuppeq0 6830 . . . . . . . 8  |-  ( ( ( B  X.  { (/)
} )  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  -> 
( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
1916, 3, 17, 18syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( ( ( B  X.  { (/) } ) supp  (/) )  =  (/)  <->  ( B  X.  { (/) } )  =  ( B  X.  { (/)
} ) ) )
2013, 19mpbird 232 . . . . . 6  |-  ( ph  ->  ( ( B  X.  { (/) } ) supp  (/) )  =  (/) )
21 oieq2 7841 . . . . . 6  |-  ( ( ( B  X.  { (/)
} ) supp  (/) )  =  (/)  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2220, 21syl 16 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( ( B  X.  { (/)
} ) supp  (/) ) )  = OrdIso (  _E  ,  (/) ) )
2322dmeqd 5153 . . . 4  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  dom OrdIso (  _E  ,  (/) ) )
24 we0 4826 . . . . . 6  |-  _E  We  (/)
25 eqid 2454 . . . . . . 7  |- OrdIso (  _E  ,  (/) )  = OrdIso (  _E  ,  (/) )
2625oien 7866 . . . . . 6  |-  ( (
(/)  e.  _V  /\  _E  We  (/) )  ->  dom OrdIso (  _E  ,  (/) )  ~~  (/) )
2714, 24, 26mp2an 672 . . . . 5  |-  dom OrdIso (  _E  ,  (/) )  ~~  (/)
28 en0 7485 . . . . 5  |-  ( dom OrdIso (  _E  ,  (/) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (/) )  =  (/) )
2927, 28mpbi 208 . . . 4  |-  dom OrdIso (  _E  ,  (/) )  =  (/)
3023, 29syl6eq 2511 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( ( B  X.  { (/) } ) supp  (/) ) )  =  (/) )
3130fveq2d 5806 . 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 7024 . . 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 2499 1  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748   {csn 3988   class class class wbr 4403    _E cep 4741    We wwe 4789   Oncon0 4830    X. cxp 4949   dom cdm 4951    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   supp csupp 6803  seq𝜔cseqom 7015    +o coa 7030    .o comu 7031    ^o coe 7032    ~~ cen 7420   finSupp cfsupp 7734  OrdIsocoi 7837   CNF ccnf 7981
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-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-map 7329  df-en 7424  df-fin 7427  df-fsupp 7735  df-oi 7838  df-cnf 7982
This theorem is referenced by:  cnfcom2lem  8048  cnfcom2lemOLD  8056
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