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Theorem oelim 8540
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 6428 . . 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 8439 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
54ad2antlr 724 . . 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 8521 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝐵))
7 onelon 6389 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 oevn0 8521 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
97, 8sylanl2 678 . . . . . . . . 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 5021 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → 𝑥𝐵 (𝐴o 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 ·o 𝐴)), 1o)‘𝑥))
146, 13eqeq12d 2747 . . . 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 718 . . 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 678 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑥𝐵 (𝐴o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  c0 4322   ciun 4997  cmpt 5231  Oncon0 6364  Lim wlim 6365  cfv 6543  (class class class)co 7412  reccrdg 8415  1oc1o 8465   ·o comu 8470  o coe 8471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oexp 8478
This theorem is referenced by:  oecl  8543  oe1m  8551  oen0  8592  oeordi  8593  oewordri  8598  oeworde  8599  oelim2  8601  oeoalem  8602  oeoelem  8604  oeeulem  8607  oe0suclim  42490  nnoeomeqom  42525
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