Theorem oelim 3137

Description: Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oelim	⊢ (((A ∈ On ∧ (B ∈ C ∧ Lim B)) ∧ ∅ ∈ A) → (A ↑o B) = ∪x ∈ B (A ↑o x))

Distinct variable group(s):   x,A   x,B

Proof of Theorem oelim

Step	Hyp	Ref	Expression
1		rdglim2a 2988	. . . 4 ⊢ ((B ∈ On ∧ Lim B) → (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x))
2	1	ad2antlr 321	. . 3 ⊢ (((A ∈ On ∧ (B ∈ On ∧ Lim B)) ∧ ∅ ∈ A) → (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x))
3		oevn0 3123	. . . . 5 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (A ↑o B) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘B))
4		oevn0 3123	. . . . . . . . . 10 ⊢ (((A ∈ On ∧ x ∈ On) ∧ ∅ ∈ A) → (A ↑o x) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x))
5		onelon 2223	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ x ∈ B) → x ∈ On)
6	4, 5	sylan12 355	. . . . . . . . 9 ⊢ (((A ∈ On ∧ (B ∈ On ∧ x ∈ B)) ∧ ∅ ∈ A) → (A ↑o x) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x))
7	6	exp42 300	. . . . . . . 8 ⊢ (A ∈ On → (B ∈ On → (x ∈ B → (∅ ∈ A → (A ↑o x) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x)))))
8	7	com34 36	. . . . . . 7 ⊢ (A ∈ On → (B ∈ On → (∅ ∈ A → (x ∈ B → (A ↑o x) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x)))))
9	8	imp31 280	. . . . . 6 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (x ∈ B → (A ↑o x) = (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x)))
10	9	iuneq2dv 2010	. . . . 5 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ∪x ∈ B (A ↑o x) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x))
11	3, 10	cleq12d 1115	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ((A ↑o B) = ∪x ∈ B (A ↑o x) ↔ (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x)))
12	11	adantlrr 315	. . 3 ⊢ (((A ∈ On ∧ (B ∈ On ∧ Lim B)) ∧ ∅ ∈ A) → ((A ↑o B) = ∪x ∈ B (A ↑o x) ↔ (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y ·o A)}, 1o) ‘x)))
13	2, 12	mpbird 171	. 2 ⊢ (((A ∈ On ∧ (B ∈ On ∧ Lim B)) ∧ ∅ ∈ A) → (A ↑o B) = ∪x ∈ B (A ↑o x))
14		limelon 2286	. . 3 ⊢ ((B ∈ C ∧ Lim B) → B ∈ On)
15		pm3.27 260	. . 3 ⊢ ((B ∈ C ∧ Lim B) → Lim B)
16	14, 15	jca 236	. 2 ⊢ ((B ∈ C ∧ Lim B) → (B ∈ On ∧ Lim B))
17	13, 16	sylan12 355	1 ⊢ (((A ∈ On ∧ (B ∈ C ∧ Lim B)) ∧ ∅ ∈ A) → (A ↑o B) = ∪x ∈ B (A ↑o x))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ∪ciun 1994  {copab 2055  Oncon0 2199  Lim wlim 2200   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem is referenced by:  oecl 3140  oe1m 3147  oen0 3165

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oexp 3108

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