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Theorem oecl 6781
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 6089 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6764 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  =  1o
3 1on 6731 . . . . . . . . 9  |-  1o  e.  On
42, 3eqeltri 2506 . . . . . . . 8  |-  ( (/)  ^o  (/) )  e.  On
51, 4syl6eqel 2524 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
65adantl 453 . . . . . 6  |-  ( ( B  e.  On  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
7 oe0m1 6765 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
87biimpa 471 . . . . . . . 8  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
9 0elon 4634 . . . . . . . 8  |-  (/)  e.  On
108, 9syl6eqel 2524 . . . . . . 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 6757 . . . . 5  |-  ( ( B  e.  On  /\  B  e.  On )  ->  ( (/)  ^o  B )  e.  On )
1312anidms 627 . . . 4  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
14 oveq1 6088 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
1514eleq1d 2502 . . . 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 6089 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  ^o  x )  =  ( A  ^o  (/) ) )
1918eleq1d 2502 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  ^o  x )  e.  On  <->  ( A  ^o  (/) )  e.  On ) )
20 oveq2 6089 . . . . . . 7  |-  ( x  =  y  ->  ( A  ^o  x )  =  ( A  ^o  y
) )
2120eleq1d 2502 . . . . . 6  |-  ( x  =  y  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  y )  e.  On ) )
22 oveq2 6089 . . . . . . 7  |-  ( x  =  suc  y  -> 
( A  ^o  x
)  =  ( A  ^o  suc  y ) )
2322eleq1d 2502 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  ^o  x )  e.  On  <->  ( A  ^o  suc  y
)  e.  On ) )
24 oveq2 6089 . . . . . . 7  |-  ( x  =  B  ->  ( A  ^o  x )  =  ( A  ^o  B
) )
2524eleq1d 2502 . . . . . 6  |-  ( x  =  B  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  B )  e.  On ) )
26 oe0 6766 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
2726, 3syl6eqel 2524 . . . . . . 7  |-  ( A  e.  On  ->  ( A  ^o  (/) )  e.  On )
2827adantr 452 . . . . . 6  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  e.  On )
29 omcl 6780 . . . . . . . . . . 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 6771 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  ^o  suc  y )  =  ( ( A  ^o  y
)  .o  A ) )
3332eleq1d 2502 . . . . . . . . 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 2959 . . . . . . . . 9  |-  x  e. 
_V
38 iunon 6600 . . . . . . . . 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 6778 . . . . . . . . . . . 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 2502 . . . . . . . 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 4844 . . . . 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 6757 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 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   (/)c0 3628   U_ciun 4093   Oncon0 4581   Lim wlim 4582   suc csuc 4583  (class class class)co 6081   1oc1o 6717    .o comu 6722    ^o coe 6723
This theorem is referenced by:  oen0  6829  oeordi  6830  oeord  6831  oecan  6832  oeword  6833  oewordri  6835  oeworde  6836  oeordsuc  6837  oeoalem  6839  oeoa  6840  oeoelem  6841  oeoe  6842  oelimcl  6843  oeeulem  6844  oeeui  6845  oaabs2  6888  omabs  6890  cantnfle  7626  cantnflt  7627  cantnfp1  7637  cantnflem1d  7644  cantnflem1  7645  cantnflem2  7646  cantnflem3  7647  cantnflem4  7648  cantnf  7649  oemapwe  7650  cantnffval2  7651  cnfcomlem  7656  cnfcom  7657  cnfcom3lem  7660  cnfcom3  7661  infxpenc  7899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-oexp 6730
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