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Theorem oe0suclim 41960
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 8517, oesuc 8522, oe0m1 8516, and oelim 8529. (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 8517 . 2 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2 oesuc 8522 . 2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
3 oelim 8529 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = 𝑐𝐵 (𝐴o 𝑐))
4 simpr 486 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∅ ∈ 𝐴)
54iftrued 4535 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = 𝑐𝐵 (𝐴o 𝑐))
63, 5eqtr4d 2776 . . . 4 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
7 simpl 484 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → 𝐴 ∈ On)
8 0elon 6415 . . . . . . . 8 ∅ ∈ On
9 ontri1 6395 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10 ss0 4397 . . . . . . . . 9 (𝐴 ⊆ ∅ → 𝐴 = ∅)
119, 10syl6bir 254 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ ∈ On) → (¬ ∅ ∈ 𝐴𝐴 = ∅))
127, 8, 11sylancl 587 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴𝐴 = ∅))
13 oveq1 7411 . . . . . . . . 9 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
14 oe0m1 8516 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1514biimpd 228 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16 0ellim 6424 . . . . . . . . . . 11 (Lim 𝐵 → ∅ ∈ 𝐵)
1715, 16impel 507 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∅)
1817adantl 483 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (∅ ↑o 𝐵) = ∅)
1913, 18sylan9eqr 2795 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ∅)
2019ex 414 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 = ∅ → (𝐴o 𝐵) = ∅))
2112, 20syld 47 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → (𝐴o 𝐵) = ∅))
2221imp 408 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴o 𝐵) = ∅)
23 simpr 486 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → ¬ ∅ ∈ 𝐴)
2423iffalsed 4538 . . . . 5 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅) = ∅)
2522, 24eqtr4d 2776 . . . 4 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
266, 25pm2.61dan 812 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
2726anassrs 469 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴o 𝐵) = if(∅ ∈ 𝐴, 𝑐𝐵 (𝐴o 𝑐), ∅))
281, 2, 27onov0suclim 41957 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 397  w3a 1088   = wceq 1542  wcel 2107  wss 3947  c0 4321  ifcif 4527   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-omul 8466  df-oexp 8467
This theorem is referenced by: (None)
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