MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnf0 Structured version   Unicode version

Theorem cantnf0 7632
Description: The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnf0.1  |-  ( 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.1 . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.2 . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2438 . . 3  |- OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) )  = OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) )
5 cantnf0.1 . . . . 5  |-  ( ph  -> 
(/)  e.  A )
6 fconst6g 5634 . . . . 5  |-  ( (/)  e.  A  ->  ( B  X.  { (/) } ) : B --> A )
75, 6syl 16 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> A )
8 0lt1o 6750 . . . . . . 7  |-  (/)  e.  1o
9 fconst6g 5634 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( B  X.  { (/) } ) : B --> 1o )
108, 9mp1i 12 . . . . . 6  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> 1o )
11 disjdif 3702 . . . . . 6  |-  ( 1o 
i^i  ( _V  \  1o ) )  =  (/)
12 fimacnvdisj 5623 . . . . . 6  |-  ( ( ( B  X.  { (/)
} ) : B --> 1o  /\  ( 1o  i^i  ( _V  \  1o ) )  =  (/) )  -> 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  =  (/) )
1310, 11, 12sylancl 645 . . . . 5  |-  ( ph  ->  ( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  =  (/) )
14 0fin 7338 . . . . 5  |-  (/)  e.  Fin
1513, 14syl6eqel 2526 . . . 4  |-  ( ph  ->  ( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  e.  Fin )
161, 2, 3cantnfs 7623 . . . 4  |-  ( ph  ->  ( ( B  X.  { (/) } )  e.  S  <->  ( ( B  X.  { (/) } ) : B --> A  /\  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) )  e.  Fin ) ) )
177, 15, 16mpbir2and 890 . . 3  |-  ( ph  ->  ( B  X.  { (/)
} )  e.  S
)
18 eqid 2438 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
191, 2, 3, 4, 17, 18cantnfval 7625 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) ) )
20 oieq2 7484 . . . . . 6  |-  ( ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) )  =  (/)  -> OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) ) )  = OrdIso
(  _E  ,  (/) ) )
2113, 20syl 16 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) ) )  = OrdIso
(  _E  ,  (/) ) )
2221dmeqd 5074 . . . 4  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) ) )  =  dom OrdIso (  _E  ,  (/) ) )
23 0ex 4341 . . . . . 6  |-  (/)  e.  _V
24 we0 4579 . . . . . 6  |-  _E  We  (/)
25 eqid 2438 . . . . . . 7  |- OrdIso (  _E  ,  (/) )  = OrdIso (  _E  ,  (/) )
2625oien 7509 . . . . . 6  |-  ( (
(/)  e.  _V  /\  _E  We  (/) )  ->  dom OrdIso (  _E  ,  (/) )  ~~  (/) )
2723, 24, 26mp2an 655 . . . . 5  |-  dom OrdIso (  _E  ,  (/) )  ~~  (/)
28 en0 7172 . . . . 5  |-  ( dom OrdIso (  _E  ,  (/) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (/) )  =  (/) )
2927, 28mpbi 201 . . . 4  |-  dom OrdIso (  _E  ,  (/) )  =  (/)
3022, 29syl6eq 2486 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) ) )  =  (/) )
3130fveq2d 5734 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) )  =  (seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) ) )
3218seqom0g 6715 . . 3  |-  ( (/)  e.  _V  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
3323, 32mp1i 12 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
3419, 31, 333eqtrd 2474 1  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    i^i cin 3321   (/)c0 3630   {csn 3816   class class class wbr 4214    _E cep 4494    We wwe 4542   Oncon0 4583    X. cxp 4878   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085  seq𝜔cseqom 6706   1oc1o 6719    +o coa 6723    .o comu 6724    ^o coe 6725    ~~ cen 7108   Fincfn 7111  OrdIsocoi 7480   CNF ccnf 7618
This theorem is referenced by:  cnfcom2lem  7660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-map 7022  df-en 7112  df-fin 7115  df-oi 7481  df-cnf 7619
  Copyright terms: Public domain W3C validator