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Theorem oe0suclim 43290
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 8560, oesuc 8565, oe0m1 8559, and oelim 8572. (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 8560 . 2 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2 oesuc 8565 . 2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
3 oelim 8572 . . . . 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 4533 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = 𝑐𝐵 (𝐴o 𝑐))
63, 5eqtr4d 2780 . . . 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 6438 . . . . . . . 8 ∅ ∈ On
9 ontri1 6418 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10 ss0 4402 . . . . . . . . 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 7438 . . . . . . . . 9 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
14 oe0m1 8559 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1514biimpd 229 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16 0ellim 6447 . . . . . . . . . . 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 2799 . . . . . . . 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 4536 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = ∅)
2522, 24eqtr4d 2780 . . . 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 43287 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 1087   = wceq 1540  wcel 2108  wss 3951  c0 4333  ifcif 4525   ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  1oc1o 8499   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-omul 8511  df-oexp 8512
This theorem is referenced by: (None)
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