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Theorem oemapwe 8017
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.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( 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.a . . . . 5  |-  ( ph  ->  A  e.  On )
2 cantnfs.b . . . . 5  |-  ( ph  ->  B  e.  On )
3 oecl 7090 . . . . 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 4840 . . . 4  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
6 ordwe 4843 . . . 4  |-  ( Ord  ( A  ^o  B
)  ->  _E  We  ( A  ^o  B ) )
74, 5, 63syl 20 . . 3  |-  ( ph  ->  _E  We  ( A  ^o  B ) )
8 cantnfs.s . . . . 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 8016 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
11 isowe 6152 . . . 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 6133 . . . . . 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 6228 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
1817dmex 6624 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
198, 18eqeltri 2538 . . . . . . 7  |-  S  e. 
_V
20 exse 4795 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2119, 20ax-mp 5 . . . . . 6  |-  T Se  S
22 eqid 2454 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2322oieu 7868 . . . . . 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 2462 . 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 2799   E.wrex 2800   _Vcvv 3078   {copab 4460    _E cep 4741   Se wse 4788    We wwe 4789   Ord word 4829   Oncon0 4830   `'ccnv 4950   dom cdm 4951   ` cfv 5529    Isom wiso 5530  (class class class)co 6203    ^o coe 7032  OrdIsocoi 7838   CNF ccnf 7982
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-int 4240  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-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-cnf 7983
This theorem is referenced by:  cantnffval2  8018  cantnffval2OLD  8040  wemapwe  8043  wemapweOLD  8044
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