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Theorem oevn0 8553
Description: Value of ordinal exponentiation at a nonzero base. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oevn0 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oevn0
StepHypRef Expression
1 on0eln0 6440 . . . . 5 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
2 df-ne 2941 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
31, 2bitrdi 287 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
43adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
5 oev 8552 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
6 iffalse 4534 . . . . 5 𝐴 = ∅ → if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
75, 6sylan9eq 2797 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ¬ 𝐴 = ∅) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
87ex 412 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 = ∅ → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
94, 8sylbid 240 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
109imp 406 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  c0 4333  ifcif 4525  cmpt 5225  Oncon0 6384  cfv 6561  (class class class)co 7431  reccrdg 8449  1oc1o 8499   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oexp 8512
This theorem is referenced by:  oe0  8560  oev2  8561  oesuclem  8563  oelim  8572
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