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Theorem oemapweOLD 8022
Description: The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of oemapwe 8000 as of 2-Jul-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 )
oemapvalOLD.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
Assertion
Ref Expression
oemapweOLD  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
Distinct variable groups:    x, w, y, z, B    w, A, x, y, z    x, S, y, z    ph, x, y, z
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)

Proof of Theorem oemapweOLD
StepHypRef Expression
1 cantnfsOLD.2 . . . . 5  |-  ( ph  ->  A  e.  On )
2 cantnfsOLD.3 . . . . 5  |-  ( ph  ->  B  e.  On )
3 oecl 7074 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 661 . . . 4  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 eloni 4824 . . . 4  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
6 ordwe 4827 . . . 4  |-  ( Ord  ( A  ^o  B
)  ->  _E  We  ( A  ^o  B ) )
74, 5, 63syl 20 . . 3  |-  ( ph  ->  _E  We  ( A  ^o  B ) )
8 cantnfsOLD.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
9 oemapvalOLD.t . . . . 5  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
108, 1, 2, 9cantnfOLD 8021 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
11 isowe 6136 . . . 4  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  -> 
( T  We  S  <->  _E  We  ( A  ^o  B ) ) )
1210, 11syl 16 . . 3  |-  ( ph  ->  ( T  We  S  <->  _E  We  ( A  ^o  B ) ) )
137, 12mpbird 232 . 2  |-  ( ph  ->  T  We  S )
144, 5syl 16 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 6117 . . . . . 6  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
1610, 15syl 16 . . . . 5  |-  ( ph  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
17 ovex 6212 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
1817dmex 6608 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
198, 18eqeltri 2533 . . . . . . 7  |-  S  e. 
_V
20 exse 4779 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2119, 20ax-mp 5 . . . . . 6  |-  T Se  S
22 eqid 2451 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2322oieu 7851 . . . . . 6  |-  ( ( T  We  S  /\  T Se  S )  ->  (
( Ord  ( A  ^o  B )  /\  `' ( A CNF  B )  Isom  _E  ,  T  ( ( A  ^o  B
) ,  S ) )  <->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF  B )  = OrdIso ( T ,  S
) ) ) )
2413, 21, 23sylancl 662 . . . . 5  |-  ( ph  ->  ( ( Ord  ( A  ^o  B )  /\  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )  <->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF  B )  = OrdIso ( T ,  S
) ) ) )
2514, 16, 24mpbi2and 912 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2625simpld 459 . . 3  |-  ( ph  ->  ( A  ^o  B
)  =  dom OrdIso ( T ,  S ) )
2726eqcomd 2458 . 2  |-  ( ph  ->  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) )
2813, 27jca 532 1  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793   E.wrex 2794   _Vcvv 3065   {copab 4444    _E cep 4725   Se wse 4772    We wwe 4773   Ord word 4813   Oncon0 4814   `'ccnv 4934   dom cdm 4935   ` cfv 5513    Isom wiso 5514  (class class class)co 6187    ^o coe 7016  OrdIsocoi 7821   CNF ccnf 7965
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-supp 6788  df-recs 6929  df-rdg 6963  df-seqom 7000  df-1o 7017  df-2o 7018  df-oadd 7021  df-omul 7022  df-oexp 7023  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-fsupp 7719  df-oi 7822  df-cnf 7966
This theorem is referenced by: (None)
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