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

Theorem oecl 6740
Description: Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
oecl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )

Proof of Theorem oecl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6048 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6723 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  =  1o
3 1on 6690 . . . . . . . . 9  |-  1o  e.  On
42, 3eqeltri 2474 . . . . . . . 8  |-  ( (/)  ^o  (/) )  e.  On
51, 4syl6eqel 2492 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
65adantl 453 . . . . . 6  |-  ( ( B  e.  On  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
7 oe0m1 6724 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
87biimpa 471 . . . . . . . 8  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
9 0elon 4594 . . . . . . . 8  |-  (/)  e.  On
108, 9syl6eqel 2492 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
1110adantll 695 . . . . . 6  |-  ( ( ( B  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
126, 11oe0lem 6716 . . . . 5  |-  ( ( B  e.  On  /\  B  e.  On )  ->  ( (/)  ^o  B )  e.  On )
1312anidms 627 . . . 4  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
14 oveq1 6047 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
1514eleq1d 2470 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  e.  On  <->  ( (/)  ^o  B
)  e.  On ) )
1613, 15syl5ibr 213 . . 3  |-  ( A  =  (/)  ->  ( B  e.  On  ->  ( A  ^o  B )  e.  On ) )
1716impcom 420 . 2  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  e.  On )
18 oveq2 6048 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  ^o  x )  =  ( A  ^o  (/) ) )
1918eleq1d 2470 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  ^o  x )  e.  On  <->  ( A  ^o  (/) )  e.  On ) )
20 oveq2 6048 . . . . . . 7  |-  ( x  =  y  ->  ( A  ^o  x )  =  ( A  ^o  y
) )
2120eleq1d 2470 . . . . . 6  |-  ( x  =  y  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  y )  e.  On ) )
22 oveq2 6048 . . . . . . 7  |-  ( x  =  suc  y  -> 
( A  ^o  x
)  =  ( A  ^o  suc  y ) )
2322eleq1d 2470 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  ^o  x )  e.  On  <->  ( A  ^o  suc  y
)  e.  On ) )
24 oveq2 6048 . . . . . . 7  |-  ( x  =  B  ->  ( A  ^o  x )  =  ( A  ^o  B
) )
2524eleq1d 2470 . . . . . 6  |-  ( x  =  B  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  B )  e.  On ) )
26 oe0 6725 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
2726, 3syl6eqel 2492 . . . . . . 7  |-  ( A  e.  On  ->  ( A  ^o  (/) )  e.  On )
2827adantr 452 . . . . . 6  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  e.  On )
29 omcl 6739 . . . . . . . . . . 11  |-  ( ( ( A  ^o  y
)  e.  On  /\  A  e.  On )  ->  ( ( A  ^o  y )  .o  A
)  e.  On )
3029expcom 425 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( A  ^o  y
)  e.  On  ->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3130adantr 452 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( ( A  ^o  y )  .o  A
)  e.  On ) )
32 oesuc 6730 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  ^o  suc  y )  =  ( ( A  ^o  y
)  .o  A ) )
3332eleq1d 2470 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  suc  y )  e.  On  <->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3431, 33sylibrd 226 . . . . . . . 8  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) )
3534expcom 425 . . . . . . 7  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  ^o  y
)  e.  On  ->  ( A  ^o  suc  y
)  e.  On ) ) )
3635adantrd 455 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) ) )
37 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
38 iunon 6559 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  ^o  y )  e.  On )  ->  U_ y  e.  x  ( A  ^o  y
)  e.  On )
3937, 38mpan 652 . . . . . . . 8  |-  ( A. y  e.  x  ( A  ^o  y )  e.  On  ->  U_ y  e.  x  ( A  ^o  y )  e.  On )
40 oelim 6737 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  U_ y  e.  x  ( A  ^o  y ) )
4137, 40mpanlr1 668 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\ 
Lim  x )  /\  (/) 
e.  A )  -> 
( A  ^o  x
)  =  U_ y  e.  x  ( A  ^o  y ) )
4241anasss 629 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( Lim  x  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4342an12s 777 . . . . . . . . 9  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4443eleq1d 2470 . . . . . . . 8  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  (
( A  ^o  x
)  e.  On  <->  U_ y  e.  x  ( A  ^o  y )  e.  On ) )
4539, 44syl5ibr 213 . . . . . . 7  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A. y  e.  x  ( A  ^o  y
)  e.  On  ->  ( A  ^o  x )  e.  On ) )
4645ex 424 . . . . . 6  |-  ( Lim  x  ->  ( ( A  e.  On  /\  (/)  e.  A
)  ->  ( A. y  e.  x  ( A  ^o  y )  e.  On  ->  ( A  ^o  x )  e.  On ) ) )
4719, 21, 23, 25, 28, 36, 46tfinds3 4803 . . . . 5  |-  ( B  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( A  ^o  B
)  e.  On ) )
4847exp3a 426 . . . 4  |-  ( B  e.  On  ->  ( A  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
4948com12 29 . . 3  |-  ( A  e.  On  ->  ( B  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
5049imp31 422 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  e.  On )
5117, 50oe0lem 6716 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   (/)c0 3588   U_ciun 4053   Oncon0 4541   Lim wlim 4542   suc csuc 4543  (class class class)co 6040   1oc1o 6676    .o comu 6681    ^o coe 6682
This theorem is referenced by:  oen0  6788  oeordi  6789  oeord  6790  oecan  6791  oeword  6792  oewordri  6794  oeworde  6795  oeordsuc  6796  oeoalem  6798  oeoa  6799  oeoelem  6800  oeoe  6801  oelimcl  6802  oeeulem  6803  oeeui  6804  oaabs2  6847  omabs  6849  cantnfle  7582  cantnflt  7583  cantnfp1  7593  cantnflem1d  7600  cantnflem1  7601  cantnflem2  7602  cantnflem3  7603  cantnflem4  7604  cantnf  7605  oemapwe  7606  cantnffval2  7607  cnfcomlem  7612  cnfcom  7613  cnfcom3lem  7616  cnfcom3  7617  infxpenc  7855
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-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-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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-oexp 6689
  Copyright terms: Public domain W3C validator