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Theorem oesuc 8528
Description: Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oesuc ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))

Proof of Theorem oesuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 limon 7821 . 2 Lim On
2 rdgsuc 8425 . 2 (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
31, 2oesuclem 8526 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cmpt 5224  Oncon0 6358  suc csuc 6360  (class class class)co 7405  1oc1o 8460   ·o comu 8465  o coe 8466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-omul 8472  df-oexp 8473
This theorem is referenced by:  oecl  8538  oe1m  8546  oen0  8587  oeordi  8588  oewordri  8593  oeordsuc  8595  oeoalem  8597  oeoelem  8599  oeeui  8603  oaabs2  8650  omabs  8652  cantnflt  9669  cnfcom  9697  infxpenc2  10019  onexoegt  42574  oe0suclim  42608  oaomoencom  42648  cantnftermord  42651  oe2  42738
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