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Theorem oeeu 8641
Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
oeeu ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem oeeu
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}
21oeeulem 8639 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ On ∧ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ⊆ 𝐵𝐵 ∈ (𝐴o suc {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})))
32simp1d 1143 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ On)
4 fvexd 6921 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∈ V)
5 fvexd 6921 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∈ V)
6 eqid 2737 . . . 4 (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)) = (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))
7 eqid 2737 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)))
8 eqid 2737 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)))
91, 6, 7, 8oeeui 8640 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ (𝑥 = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴o {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))))))
103, 4, 5, 9euotd 5518 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
11 df-3an 1089 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ 𝑧 ∈ (𝐴o 𝑥)))
1211biancomi 462 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ (𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))))
1312anbi1i 624 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
1413anbi2i 623 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
15 an12 645 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
16 anass 468 . . . . . . . 8 (((𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
1714, 15, 163bitri 297 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
1817exbii 1848 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
19 df-rex 3071 . . . . . 6 (∃𝑧 ∈ (𝐴o 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
20 r19.42v 3191 . . . . . 6 (∃𝑧 ∈ (𝐴o 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
2118, 19, 203bitr2i 299 . . . . 5 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
22212exbii 1849 . . . 4 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
23 r2ex 3196 . . . 4 (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
2422, 23bitr4i 278 . . 3 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
2524eubii 2585 . 2 (∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
2610, 25sylib 218 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  ∃!weu 2568  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  cop 4632  cotp 4634   cuni 4907   cint 4946  Oncon0 6384  suc csuc 6386  cio 6512  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  2oc2o 8500   +o coa 8503   ·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-rmo 3380  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-ot 4635  df-uni 4908  df-int 4947  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-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by:  onexoegt  43256  omabs2  43345
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