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Theorem oev 8552
Description: Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
oev ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2741 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 oveq2 7439 . . . . . 6 (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
32mpteq2dv 5244 . . . . 5 (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
4 rdgeq1 8451 . . . . 5 ((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
53, 4syl 17 . . . 4 (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
65fveq1d 6908 . . 3 (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
71, 6ifbieq2d 4552 . 2 (𝑦 = 𝐴 → if(𝑦 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)))
8 difeq2 4120 . . 3 (𝑧 = 𝐵 → (1o𝑧) = (1o𝐵))
9 fveq2 6906 . . 3 (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
108, 9ifeq12d 4547 . 2 (𝑧 = 𝐵 → if(𝐴 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
11 df-oexp 8512 . 2 o = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)))
12 1oex 8516 . . . 4 1o ∈ V
1312difexi 5330 . . 3 (1o𝐵) ∈ V
14 fvex 6919 . . 3 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V
1513, 14ifex 4576 . 2 if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) ∈ V
167, 10, 11, 15ovmpo 7593 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  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-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-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-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:  oevn0  8553  oe0m  8556
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