MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflt2OLD Structured version   Unicode version

Theorem cantnflt2OLD 8111
Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnflt2 8081 as of 29-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
cantnflt2OLD.4  |-  ( ph  ->  F  e.  S )
cantnflt2OLD.5  |-  ( ph  -> 
(/)  e.  A )
cantnflt2OLD.6  |-  ( ph  ->  C  e.  On )
cantnflt2OLD.7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
Assertion
Ref Expression
cantnflt2OLD  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )

Proof of Theorem cantnflt2OLD
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfsOLD.1 . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfsOLD.2 . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.3 . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2460 . . 3  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
5 cantnflt2OLD.4 . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2460 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
71, 2, 3, 4, 5, 6cantnfvalOLD 8106 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) ) )
8 cantnflt2OLD.5 . . 3  |-  ( ph  -> 
(/)  e.  A )
9 cantnflt2OLD.6 . . . . 5  |-  ( ph  ->  C  e.  On )
10 cantnflt2OLD.7 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
119, 10ssexd 4587 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
124oion 7950 . . . 4  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On )
13 sucidg 4949 . . . 4  |-  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  suc  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )
1411, 12, 133syl 20 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
suc  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) )
151, 2, 3, 4, 5cantnfclOLD 8105 . . . . . . 7  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
om ) )
1615simpld 459 . . . . . 6  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
174oiiso 7951 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  -> OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) ,  ( `' F " ( _V 
\  1o ) ) ) )
1811, 16, 17syl2anc 661 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) ,  ( `' F "
( _V  \  1o ) ) ) )
19 isof1o 6200 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) ,  ( `' F " ( _V 
\  1o ) ) )  -> OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) ) )
20 f1ofo 5814 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) )  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) : dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) -onto-> ( `' F " ( _V 
\  1o ) ) )
21 foima 5791 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -onto-> ( `' F " ( _V 
\  1o ) )  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2218, 19, 20, 214syl 21 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2322, 10eqsstrd 3531 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  C_  C
)
241, 2, 3, 4, 5, 6, 8, 14, 9, 23cantnfltOLD 8110 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) )  e.  ( A  ^o  C ) )
257, 24eqeltrd 2548 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    C_ wss 3469   (/)c0 3778    _E cep 4782    We wwe 4830   Oncon0 4871   suc csuc 4873   `'ccnv 4991   dom cdm 4992   "cima 4995   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579    Isom wiso 5580  (class class class)co 6275    |-> cmpt2 6277   omcom 6671  seq𝜔cseqom 7102   1oc1o 7113    +o coa 7117    .o comu 7118    ^o coe 7119  OrdIsocoi 7923   CNF ccnf 8067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-seqom 7103  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-oexp 7126  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-cnf 8068
This theorem is referenced by:  cantnflem1dOLD  8119  cnfcom3lemOLD  8144
  Copyright terms: Public domain W3C validator