Theorem omlim 3136

Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62.

Assertion

Ref	Expression
omlim	⊢ ((A ∈ On ∧ (B ∈ C ∧ Lim B)) → (A ·o B) = ∪x ∈ B (A ·o x))

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

Proof of Theorem omlim

Step	Hyp	Ref	Expression
1		rdglim2a 2988	. . . 4 ⊢ ((B ∈ On ∧ Lim B) → (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
2	1	adantl 305	. . 3 ⊢ ((A ∈ On ∧ (B ∈ On ∧ Lim B)) → (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
3		omv 3120	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → (A ·o B) = (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘B))
4		omv 3120	. . . . . . . . 9 ⊢ ((A ∈ On ∧ x ∈ On) → (A ·o x) = (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
5		onelon 2223	. . . . . . . . 9 ⊢ ((B ∈ On ∧ x ∈ B) → x ∈ On)
6	4, 5	sylan2 346	. . . . . . . 8 ⊢ ((A ∈ On ∧ (B ∈ On ∧ x ∈ B)) → (A ·o x) = (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
7	6	anassrs 338	. . . . . . 7 ⊢ (((A ∈ On ∧ B ∈ On) ∧ x ∈ B) → (A ·o x) = (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
8	7	exp 291	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (x ∈ B → (A ·o x) = (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x)))
9	8	iuneq2dv 2010	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → ∪x ∈ B (A ·o x) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x))
10	3, 9	cleq12d 1115	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → ((A ·o B) = ∪x ∈ B (A ·o x) ↔ (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x)))
11	10	adantrr 312	. . 3 ⊢ ((A ∈ On ∧ (B ∈ On ∧ Lim B)) → ((A ·o B) = ∪x ∈ B (A ·o x) ↔ (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘B) = ∪x ∈ B (rec({⟨y, z⟩∣z = (y +o A)}, ∅) ‘x)))
12	2, 11	mpbird 171	. 2 ⊢ ((A ∈ On ∧ (B ∈ On ∧ Lim B)) → (A ·o B) = ∪x ∈ B (A ·o x))
13		limelon 2286	. . 3 ⊢ ((B ∈ C ∧ Lim B) → B ∈ On)
14		pm3.27 260	. . 3 ⊢ ((B ∈ C ∧ Lim B) → Lim B)
15	13, 14	jca 236	. 2 ⊢ ((B ∈ C ∧ Lim B) → (B ∈ On ∧ Lim B))
16	12, 15	sylan2 346	1 ⊢ ((A ∈ On ∧ (B ∈ C ∧ Lim B)) → (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   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  omcl 3139  om0r 3142  om1r 3145  omordi 3164

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-omul 3107

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