Theorem oe0suclim 43273

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 8489, oesuc 8494, oe0m1 8488, and oelim 8501. (Contributed by RP, 18-Jan-2025.)

Assertion

Ref	Expression
oe0suclim	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) = ((𝐴 ↑o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))))

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem oe0suclim

Step	Hyp	Ref	Expression
1		oe0 8489	. 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
2		oesuc 8494	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
3		oelim 8501	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐))
4		simpr 484	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∅ ∈ 𝐴)
5	4	iftrued 4499	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐))
6	3, 5	eqtr4d 2768	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
7		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → 𝐴 ∈ On)
8		0elon 6390	. . . . . . . 8 ⊢ ∅ ∈ On
9		ontri1 6369	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10		ss0 4368	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
11	9, 10	biimtrrdi 254	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
12	7, 8, 11	sylancl 586	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
13		oveq1 7397	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
14		oe0m1 8488	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
15	14	biimpd 229	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16		0ellim 6399	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
17	15, 16	impel 505	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∅)
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (∅ ↑o 𝐵) = ∅)
19	13, 18	sylan9eqr 2787	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∅)
20	19	ex 412	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 = ∅ → (𝐴 ↑o 𝐵) = ∅))
21	12, 20	syld 47	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) = ∅))
22	21	imp 406	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∅)
23		simpr 484	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → ¬ ∅ ∈ 𝐴)
24	23	iffalsed 4502	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∅)
25	22, 24	eqtr4d 2768	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
26	6, 25	pm2.61dan 812	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
27	26	anassrs 467	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
28	1, 2, 27	onov0suclim 43270	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) = ((𝐴 ↑o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  ∅c0 4299  ifcif 4491  ∪ ciun 4958  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7390  1_oc1o 8430   ·_o comu 8435   ↑_o coe 8436

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-omul 8442  df-oexp 8443

This theorem is referenced by: (None)