Theorem oe0suclim 43705

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 8457, oesuc 8462, oe0m1 8456, and oelim 8469. (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 8457	. 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
2		oesuc 8462	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
3		oelim 8469	. . . . 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 4474	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐))
6	3, 5	eqtr4d 2774	. . . 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 6378	. . . . . . . 8 ⊢ ∅ ∈ On
9		ontri1 6357	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ⊆ ∅ ↔ ¬ ∅ ∈ 𝐴))
10		ss0 4342	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
11	9, 10	biimtrrdi 254	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
12	7, 8, 11	sylancl 587	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (¬ ∅ ∈ 𝐴 → 𝐴 = ∅))
13		oveq1 7374	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
14		oe0m1 8456	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
15	14	biimpd 229	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 → (∅ ↑o 𝐵) = ∅))
16		0ellim 6387	. . . . . . . . . . 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 2793	. . . . . . . 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 4477	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅) = ∅)
25	22, 24	eqtr4d 2774	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) ∧ ¬ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))
26	6, 25	pm2.61dan 813	. . 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 43702	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 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∅c0 4273  ifcif 4466  ∪ ciun 4933  Oncon0 6323  Lim wlim 6324  suc csuc 6325  (class class class)co 7367  1_oc1o 8398   ·_o comu 8403   ↑_o coe 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-omul 8410  df-oexp 8411

This theorem is referenced by: (None)