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

Theorem oecl 8532
Description: Closure law for ordinal exponentiation. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
oecl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)

Proof of Theorem oecl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7412 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8515 . . . . . . . . 9 (∅ ↑o ∅) = 1o
3 1on 8473 . . . . . . . . 9 1o ∈ On
42, 3eqeltri 2830 . . . . . . . 8 (∅ ↑o ∅) ∈ On
51, 4eqeltrdi 2842 . . . . . . 7 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
65adantl 483 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7 oe0m1 8516 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
87biimpa 478 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9 0elon 6415 . . . . . . . 8 ∅ ∈ On
108, 9eqeltrdi 2842 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
1110adantll 713 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
126, 11oe0lem 8508 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
1312anidms 568 . . . 4 (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14 oveq1 7411 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
1514eleq1d 2819 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
1613, 15imbitrrid 245 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴o 𝐵) ∈ On))
1716impcom 409 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) ∈ On)
18 oveq2 7412 . . . . . . 7 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
1918eleq1d 2819 . . . . . 6 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ On ↔ (𝐴o ∅) ∈ On))
20 oveq2 7412 . . . . . . 7 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
2120eleq1d 2819 . . . . . 6 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝑦) ∈ On))
22 oveq2 7412 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
2322eleq1d 2819 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o suc 𝑦) ∈ On))
24 oveq2 7412 . . . . . . 7 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
2524eleq1d 2819 . . . . . 6 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝐵) ∈ On))
26 oe0 8517 . . . . . . . 8 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2726, 3eqeltrdi 2842 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) ∈ On)
2827adantr 482 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) ∈ On)
29 omcl 8531 . . . . . . . . . . 11 (((𝐴o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝑦) ·o 𝐴) ∈ On)
3029expcom 415 . . . . . . . . . 10 (𝐴 ∈ On → ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o 𝐴) ∈ On))
3130adantr 482 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o 𝐴) ∈ On))
32 oesuc 8522 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
3332eleq1d 2819 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o suc 𝑦) ∈ On ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ On))
3431, 33sylibrd 259 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On))
3534expcom 415 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
3635adantrd 493 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
37 vex 3479 . . . . . . . . 9 𝑥 ∈ V
38 iunon 8334 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On) → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
3937, 38mpan 689 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
40 oelim 8529 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4137, 40mpanlr1 705 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4241anasss 468 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4342an12s 648 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4443eleq1d 2819 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴o 𝑦) ∈ On))
4539, 44imbitrrid 245 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On))
4645ex 414 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7849 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On))
4847expd 417 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
5049imp31 419 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On)
5117, 50oe0lem 8508 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  c0 4321   ciun 4996  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7404  1oc1o 8454   ·o comu 8459  o coe 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466  df-oexp 8467
This theorem is referenced by:  oen0  8582  oeordi  8583  oeord  8584  oecan  8585  oeword  8586  oewordri  8588  oeworde  8589  oeordsuc  8590  oeoalem  8592  oeoa  8593  oeoelem  8594  oeoe  8595  oelimcl  8596  oeeulem  8597  oeeui  8598  oaabs2  8644  omabs  8646  cantnfle  9662  cantnflt  9663  cantnfp1  9672  cantnflem1d  9679  cantnflem1  9680  cantnflem2  9681  cantnflem3  9682  cantnflem4  9683  cantnf  9684  oemapwe  9685  cantnffval2  9686  cnfcomlem  9690  cnfcom  9691  cnfcom3lem  9694  cnfcom3  9695  infxpenc  10009  onexoegt  41926  oaomoecl  41961  oenassex  42001  cantnftermord  42003  cantnfresb  42007  oacl2g  42013  omabs2  42015  omcl2  42016  ofoaf  42038  ofoafo  42039
  Copyright terms: Public domain W3C validator