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Theorem oeeu 8630
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 2728 . . . . 5 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}
21oeeulem 8628 . . . 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 1139 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ On)
4 fvexd 6917 . . 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 6917 . . 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 2728 . . . 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 2728 . . . 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 2728 . . . 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 8629 . . 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 5519 . 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 1086 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ 𝑧 ∈ (𝐴o 𝑥)))
1211biancomi 461 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴o 𝑥)) ↔ (𝑧 ∈ (𝐴o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))))
1312anbi1i 622 . . . . . . . . 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 621 . . . . . . . 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 643 . . . . . . . 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 467 . . . . . . . 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 296 . . . . . . 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 1842 . . . . . 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 3068 . . . . . 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 3188 . . . . . 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 298 . . . . 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 1843 . . . 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 3193 . . . 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 277 . . 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 2574 . 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 217 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 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2557  wrex 3067  {crab 3430  Vcvv 3473  cdif 3946  wss 3949  cop 4638  cotp 4640   cuni 4912   cint 4953  Oncon0 6374  suc csuc 6376  cio 6503  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  1oc1o 8486  2oc2o 8487   +o coa 8490   ·o comu 8491  o coe 8492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-ot 4641  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-omul 8498  df-oexp 8499
This theorem is referenced by:  onexoegt  42703  omabs2  42792
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