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Theorem oelim 7501
 Description: Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (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 5705 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 simpr 476 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → Lim 𝐵)
31, 2jca 553 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4 rdglim2a 7416 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
54ad2antlr 759 . . 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 7482 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝐵))
7 onelon 5665 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 oevn0 7482 . . . . . . . . . 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 637 . . . . . . . 8 (𝐴 ∈ On → (𝐵 ∈ On → (𝑥𝐵 → (∅ ∈ 𝐴 → (𝐴𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)))))
1110com34 89 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝑥𝐵 → (𝐴𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥)))))
1211imp41 617 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → (𝐴𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
1312iuneq2dv 4478 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → 𝑥𝐵 (𝐴𝑜 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·𝑜 𝐴)), 1𝑜)‘𝑥))
146, 13eqeq12d 2625 . . . 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 753 . . 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 246 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ∪ ciun 4455   ↦ cmpt 4643  Oncon0 5640  Lim wlim 5641  ‘cfv 5804  (class class class)co 6549  reccrdg 7392  1𝑜c1o 7440   ·𝑜 comu 7445   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oexp 7453 This theorem is referenced by:  oecl  7504  oe1m  7512  oen0  7553  oeordi  7554  oewordri  7559  oeworde  7560  oelim2  7562  oeoalem  7563  oeoelem  7565  oeeulem  7568
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