Theorem oe0suclim 43859

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 8493, oesuc 8498, oe0m1 8492, and oelim 8505. (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 8493	. 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
2		oesuc 8498	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
3		oelim 8505	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐))
4		simpr 488	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∅ ∈ 𝐴)
5	4	iftrued 4490	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐))
6	3, 5	eqtr4d 2802	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
7		simpl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → 𝐴 ∈ On)
8		0elon 6403	. . . . . . . 8 ⊢ ∅ ∈ On
9		ontri1 6382	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10		ss0 4358	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
11	9, 10	biimtrrdi 256	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
12	7, 8, 11	sylancl 595	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
13		oveq1 7405	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
14		oe0m1 8492	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
15	14	biimpd 231	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16		0ellim 6412	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
17	15, 16	impel 513	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∅)
18	17	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (∅ ↑o 𝐵) = ∅)
19	13, 18	sylan9eqr 2821	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∅)
20	19	ex 416	. . . . . . 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 410	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∅)
23		simpr 488	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → ¬ ∅ ∈ 𝐴)
24	23	iffalsed 4493	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∅)
25	22, 24	eqtr4d 2802	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
26	6, 25	pm2.61dan 822	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
27	26	anassrs 471	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
28	1, 2, 27	onov0suclim 43856	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 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ⊆ wss 3906  ∅c0 4287  ifcif 4482  ∪ ciun 4951  Oncon0 6348  Lim wlim 6349  suc csuc 6350  (class class class)co 7398  1_oc1o 8432   ·_o comu 8437   ↑_o coe 8438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-omul 8444  df-oexp 8445

This theorem is referenced by: (None)