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Theorem oe0suclim 43266
Description: Closed form expression of the value of ordinal exponentiation for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.6 of [Schloeder] p. 4. See oe0 8558, oesuc 8563, oe0m1 8557, and oelim 8570. (Contributed by RP, 18-Jan-2025.)
Assertion
Ref Expression
oe0suclim ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴o 𝐵) = ((𝐴o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐
Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem oe0suclim
StepHypRef Expression
1 oe0 8558 . 2 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2 oesuc 8563 . 2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
3 oelim 8570 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑐𝐵 (𝐴o 𝑐))
4 simpr 484 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∅ ∈ 𝐴)
54iftrued 4538 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = 𝑐𝐵 (𝐴o 𝑐))
63, 5eqtr4d 2777 . . . 4 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
7 simpl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → 𝐴 ∈ On)
8 0elon 6439 . . . . . . . 8 ∅ ∈ On
9 ontri1 6419 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10 ss0 4407 . . . . . . . . 9 (𝐴 ⊆ ∅ → 𝐴 = ∅)
119, 10biimtrrdi 254 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ ∈ On) → (¬ ∅ ∈ 𝐴𝐴 = ∅))
127, 8, 11sylancl 586 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴𝐴 = ∅))
13 oveq1 7437 . . . . . . . . 9 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
14 oe0m1 8557 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1514biimpd 229 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16 0ellim 6448 . . . . . . . . . . 11 (Lim 𝐵 → ∅ ∈ 𝐵)
1715, 16impel 505 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∅)
1817adantl 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (∅ ↑o 𝐵) = ∅)
1913, 18sylan9eqr 2796 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ∅)
2019ex 412 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 = ∅ → (𝐴o 𝐵) = ∅))
2112, 20syld 47 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → (𝐴o 𝐵) = ∅))
2221imp 406 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴o 𝐵) = ∅)
23 simpr 484 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → ¬ ∅ ∈ 𝐴)
2423iffalsed 4541 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = ∅)
2522, 24eqtr4d 2777 . . . 4 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
266, 25pm2.61dan 813 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
2726anassrs 467 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
281, 2, 27onov0suclim 43263 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴o 𝐵) = ((𝐴o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wss 3962  c0 4338  ifcif 4530   ciun 4995  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  1oc1o 8497   ·o comu 8502  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-omul 8509  df-oexp 8510
This theorem is referenced by: (None)
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