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Theorem oemapwe 7890
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 6965 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 654 . . . 4  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 eloni 4716 . . . 4  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
6 ordwe 4719 . . . 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 7889 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
11 isowe 6027 . . . 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 6008 . . . . . 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 6105 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
1817dmex 6500 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
198, 18eqeltri 2503 . . . . . . 7  |-  S  e. 
_V
20 exse 4671 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2119, 20ax-mp 5 . . . . . 6  |-  T Se  S
22 eqid 2433 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2322oieu 7741 . . . . . 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 655 . . . . 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 905 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2625simpld 456 . . 3  |-  ( ph  ->  ( A  ^o  B
)  =  dom OrdIso ( T ,  S ) )
2726eqcomd 2438 . 2  |-  ( ph  ->  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) )
2813, 27jca 529 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 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962   {copab 4337    _E cep 4617   Se wse 4664    We wwe 4665   Ord word 4705   Oncon0 4706   `'ccnv 4826   dom cdm 4827   ` cfv 5406    Isom wiso 5407  (class class class)co 6080    ^o coe 6907  OrdIsocoi 7711   CNF ccnf 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-seqom 6889  df-1o 6908  df-2o 6909  df-oadd 6912  df-omul 6913  df-oexp 6914  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-oi 7712  df-cnf 7856
This theorem is referenced by:  cantnffval2  7891  cantnffval2OLD  7913  wemapwe  7916  wemapweOLD  7917
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