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Theorem oelim 8572
Description: Ordinal exponentiation with a limit exponent and nonzero base. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oelim (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem oelim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limelon 6448 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 simpr 484 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → Lim 𝐵)
31, 2jca 511 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4 rdglim2a 8473 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
54ad2antlr 727 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
6 oevn0 8553 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵))
7 onelon 6409 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 oevn0 8553 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
97, 8sylanl2 681 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
109exp42 435 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ On → (𝑥𝐵 → (∅ ∈ 𝐴 → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
1110com34 91 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝑥𝐵 → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))))
1211imp41 425 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
1312iuneq2dv 5016 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → 𝑥𝐵 (𝐴o 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
146, 13eqeq12d 2753 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))
1514adantlrr 721 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ((𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥)))
165, 15mpbird 257 . 2 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥))
173, 16sylanl2 681 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333   ciun 4991  cmpt 5225  Oncon0 6384  Lim wlim 6385  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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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-iun 4993  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-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oexp 8512
This theorem is referenced by:  oecl  8575  oe1m  8583  oen0  8624  oeordi  8625  oewordri  8630  oeworde  8631  oelim2  8633  oeoalem  8634  oeoelem  8636  oeeulem  8639  oe0suclim  43290  nnoeomeqom  43325
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