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Theorem oecl 6536
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 5866 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6519 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  =  1o
3 1on 6486 . . . . . . . . 9  |-  1o  e.  On
42, 3eqeltri 2353 . . . . . . . 8  |-  ( (/)  ^o  (/) )  e.  On
51, 4syl6eqel 2371 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
65adantl 452 . . . . . 6  |-  ( ( B  e.  On  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
7 oe0m1 6520 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
87biimpa 470 . . . . . . . 8  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
9 0elon 4445 . . . . . . . 8  |-  (/)  e.  On
108, 9syl6eqel 2371 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
1110adantll 694 . . . . . 6  |-  ( ( ( B  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
126, 11oe0lem 6512 . . . . 5  |-  ( ( B  e.  On  /\  B  e.  On )  ->  ( (/)  ^o  B )  e.  On )
1312anidms 626 . . . 4  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
14 oveq1 5865 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
1514eleq1d 2349 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  e.  On  <->  ( (/)  ^o  B
)  e.  On ) )
1613, 15syl5ibr 212 . . 3  |-  ( A  =  (/)  ->  ( B  e.  On  ->  ( A  ^o  B )  e.  On ) )
1716impcom 419 . 2  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  e.  On )
18 oveq2 5866 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  ^o  x )  =  ( A  ^o  (/) ) )
1918eleq1d 2349 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  ^o  x )  e.  On  <->  ( A  ^o  (/) )  e.  On ) )
20 oveq2 5866 . . . . . . 7  |-  ( x  =  y  ->  ( A  ^o  x )  =  ( A  ^o  y
) )
2120eleq1d 2349 . . . . . 6  |-  ( x  =  y  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  y )  e.  On ) )
22 oveq2 5866 . . . . . . 7  |-  ( x  =  suc  y  -> 
( A  ^o  x
)  =  ( A  ^o  suc  y ) )
2322eleq1d 2349 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  ^o  x )  e.  On  <->  ( A  ^o  suc  y
)  e.  On ) )
24 oveq2 5866 . . . . . . 7  |-  ( x  =  B  ->  ( A  ^o  x )  =  ( A  ^o  B
) )
2524eleq1d 2349 . . . . . 6  |-  ( x  =  B  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  B )  e.  On ) )
26 oe0 6521 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
2726, 3syl6eqel 2371 . . . . . . 7  |-  ( A  e.  On  ->  ( A  ^o  (/) )  e.  On )
2827adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  e.  On )
29 omcl 6535 . . . . . . . . . . 11  |-  ( ( ( A  ^o  y
)  e.  On  /\  A  e.  On )  ->  ( ( A  ^o  y )  .o  A
)  e.  On )
3029expcom 424 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( A  ^o  y
)  e.  On  ->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3130adantr 451 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( ( A  ^o  y )  .o  A
)  e.  On ) )
32 oesuc 6526 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  ^o  suc  y )  =  ( ( A  ^o  y
)  .o  A ) )
3332eleq1d 2349 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  suc  y )  e.  On  <->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3431, 33sylibrd 225 . . . . . . . 8  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) )
3534expcom 424 . . . . . . 7  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  ^o  y
)  e.  On  ->  ( A  ^o  suc  y
)  e.  On ) ) )
3635adantrd 454 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) ) )
37 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
38 iunon 6355 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  ^o  y )  e.  On )  ->  U_ y  e.  x  ( A  ^o  y
)  e.  On )
3937, 38mpan 651 . . . . . . . 8  |-  ( A. y  e.  x  ( A  ^o  y )  e.  On  ->  U_ y  e.  x  ( A  ^o  y )  e.  On )
40 oelim 6533 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  U_ y  e.  x  ( A  ^o  y ) )
4137, 40mpanlr1 667 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\ 
Lim  x )  /\  (/) 
e.  A )  -> 
( A  ^o  x
)  =  U_ y  e.  x  ( A  ^o  y ) )
4241anasss 628 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( Lim  x  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4342an12s 776 . . . . . . . . 9  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4443eleq1d 2349 . . . . . . . 8  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  (
( A  ^o  x
)  e.  On  <->  U_ y  e.  x  ( A  ^o  y )  e.  On ) )
4539, 44syl5ibr 212 . . . . . . 7  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A. y  e.  x  ( A  ^o  y
)  e.  On  ->  ( A  ^o  x )  e.  On ) )
4645ex 423 . . . . . 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 4655 . . . . 5  |-  ( B  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( A  ^o  B
)  e.  On ) )
4847exp3a 425 . . . 4  |-  ( B  e.  On  ->  ( A  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
4948com12 27 . . 3  |-  ( A  e.  On  ->  ( B  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
5049imp31 421 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  e.  On )
5117, 50oe0lem 6512 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   (/)c0 3455   U_ciun 3905   Oncon0 4392   Lim wlim 4393   suc csuc 4394  (class class class)co 5858   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem is referenced by:  oen0  6584  oeordi  6585  oeord  6586  oecan  6587  oeword  6588  oewordri  6590  oeworde  6591  oeordsuc  6592  oeoalem  6594  oeoa  6595  oeoelem  6596  oeoe  6597  oelimcl  6598  oeeulem  6599  oeeui  6600  oaabs2  6643  omabs  6645  cantnfle  7372  cantnflt  7373  cantnfp1  7383  cantnflem1d  7390  cantnflem1  7391  cantnflem2  7392  cantnflem3  7393  cantnflem4  7394  cantnf  7395  oemapwe  7396  cantnffval2  7397  cnfcomlem  7402  cnfcom  7403  cnfcom3lem  7406  cnfcom3  7407  infxpenc  7645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-oexp 6485
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