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Theorem oesuc 7175
Description: Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oesuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )

Proof of Theorem oesuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 limon 6652 . 2  |-  Lim  On
2 rdgsuc 7088 . 2  |-  ( B  e.  On  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  .o  A ) ) `  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) ) )
31, 2oesuclem 7173 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    |-> cmpt 4491   Oncon0 4864   suc csuc 4866  (class class class)co 6277   1oc1o 7121    .o comu 7126    ^o coe 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-1o 7128  df-omul 7133  df-oexp 7134
This theorem is referenced by:  oecl  7185  oe1m  7192  oen0  7233  oeordi  7234  oewordri  7239  oeordsuc  7241  oeoalem  7243  oeoelem  7245  oeeui  7249  oaabs2  7292  omabs  7294  cantnflt  8089  cantnfltOLD  8119  cnfcom  8142  cnfcomOLD  8150  infxpenc2  8397  infxpenc2OLD  8401
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