MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeeu Structured version   Visualization version   GIF version

Theorem oeeu 8550
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 2736 . . . . 5 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)}
21oeeulem 8548 . . . 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 1142 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑎)} ∈ On)
4 fvexd 6857 . . 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 6857 . . 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 2736 . . . 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 2736 . . . 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 2736 . . . 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 8549 . . 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 5470 . 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 463 . . . . . . . . . 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 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 469 . . . . . . . 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 1850 . . . . . 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 3074 . . . . . 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 3187 . . . . . 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 1851 . . . 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 3192 . . . 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 2583 . 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  cop 4592  cotp 4594   cuni 4865   cint 4907  Oncon0 6317  suc csuc 6319  cio 6446  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  1oc1o 8405  2oc2o 8406   +o coa 8409   ·o comu 8410  o coe 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  onexoegt  41564  omabs2  41651
  Copyright terms: Public domain W3C validator