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Theorem oeeu 8032
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 2778 . . . . . 6 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}
21oeeulem 8030 . . . . 5 ((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 1122 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ On)
43elexd 3435 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ V)
5 fvexd 6516 . . 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)
6 fvexd 6516 . . 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)
7 eqid 2778 . . . 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))
8 eqid 2778 . . . 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)))
9 eqid 2778 . . . 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)))
101, 7, 8, 9oeeui 8031 . . 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))))))
114, 5, 6, 10euotd 5260 . 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)))
12 df-3an 1070 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ 𝑧 ∈ (𝐴o 𝑥)))
1312biancomi 455 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ (𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))))
1413anbi1i 614 . . . . . . . . 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))
1514anbi2i 613 . . . . . . . 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)))
16 an12 632 . . . . . . . 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)))
17 anass 461 . . . . . . . 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))))
1815, 16, 173bitri 289 . . . . . . 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))))
1918exbii 1810 . . . . . 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))))
20 df-rex 3094 . . . . . 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))))
21 r19.42v 3291 . . . . . 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)))
2219, 20, 213bitr2i 291 . . . . 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)))
23222exbii 1811 . . . 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)))
24 r2ex 3248 . . . 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)))
2523, 24bitr4i 270 . . 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))
2625eubii 2603 . 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))
2711, 26sylib 210 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 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  ∃!weu 2582  wrex 3089  {crab 3092  cdif 3828  wss 3831  cop 4448  cotp 4450   cuni 4713   cint 4750  Oncon0 6031  suc csuc 6033  cio 6152  cfv 6190  (class class class)co 6978  1st c1st 7501  2nd c2nd 7502  1oc1o 7900  2oc2o 7901   +o coa 7904   ·o comu 7905  o coe 7906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-ot 4451  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-2o 7908  df-oadd 7911  df-omul 7912  df-oexp 7913
This theorem is referenced by: (None)
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