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Theorem oemapwe 7606
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.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
oemapval.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
oemapwe  |-  ( 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 oemapwe
StepHypRef Expression
1 cantnfs.2 . . . . 5  |-  ( ph  ->  A  e.  On )
2 cantnfs.3 . . . . 5  |-  ( ph  ->  B  e.  On )
3 oecl 6740 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 643 . . . 4  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 eloni 4551 . . . 4  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
6 ordwe 4554 . . . 4  |-  ( Ord  ( A  ^o  B
)  ->  _E  We  ( A  ^o  B ) )
74, 5, 63syl 19 . . 3  |-  ( ph  ->  _E  We  ( A  ^o  B ) )
8 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
9 oemapval.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, 9cantnf 7605 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
11 isowe 6028 . . . 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 224 . 2  |-  ( ph  ->  T  We  S )
144, 5syl 16 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 6009 . . . . . 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 6065 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
1817dmex 5091 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
198, 18eqeltri 2474 . . . . . . 7  |-  S  e. 
_V
20 exse 4506 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2119, 20ax-mp 8 . . . . . 6  |-  T Se  S
22 eqid 2404 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2322oieu 7464 . . . . . 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 644 . . . . 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 888 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2625simpld 446 . . 3  |-  ( ph  ->  ( A  ^o  B
)  =  dom OrdIso ( T ,  S ) )
2726eqcomd 2409 . 2  |-  ( ph  ->  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) )
2813, 27jca 519 1  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   {copab 4225    _E cep 4452   Se wse 4499    We wwe 4500   Ord word 4540   Oncon0 4541   `'ccnv 4836   dom cdm 4837   ` cfv 5413    Isom wiso 5414  (class class class)co 6040    ^o coe 6682  OrdIsocoi 7434   CNF ccnf 7572
This theorem is referenced by:  cantnffval2  7607  wemapwe  7610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-oexp 6689  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-cnf 7573
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