Theorem omcl 3139

Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57.

Assertion

Ref	Expression
omcl	⊢ ((A ∈ On ∧ B ∈ On) → (A ·o B) ∈ On)

Proof of Theorem omcl

Step	Hyp	Ref	Expression
1		opreq2 3007	. . . . 5 ⊢ (x = ∅ → (A ·o x) = (A ·o ∅))
2	1	eleq1d 1155	. . . 4 ⊢ (x = ∅ → ((A ·o x) ∈ On ↔ (A ·o ∅) ∈ On))
3		opreq2 3007	. . . . 5 ⊢ (x = y → (A ·o x) = (A ·o y))
4	3	eleq1d 1155	. . . 4 ⊢ (x = y → ((A ·o x) ∈ On ↔ (A ·o y) ∈ On))
5		opreq2 3007	. . . . 5 ⊢ (x = suc y → (A ·o x) = (A ·o suc y))
6	5	eleq1d 1155	. . . 4 ⊢ (x = suc y → ((A ·o x) ∈ On ↔ (A ·o suc y) ∈ On))
7		opreq2 3007	. . . . 5 ⊢ (x = B → (A ·o x) = (A ·o B))
8	7	eleq1d 1155	. . . 4 ⊢ (x = B → ((A ·o x) ∈ On ↔ (A ·o B) ∈ On))
9		0elon 2277	. . . . 5 ⊢ ∅ ∈ On
10		om0 3125	. . . . . 6 ⊢ (A ∈ On → (A ·o ∅) = ∅)
11	10	eleq1d 1155	. . . . 5 ⊢ (A ∈ On → ((A ·o ∅) ∈ On ↔ ∅ ∈ On))
12	9, 11	mpbiri 169	. . . 4 ⊢ (A ∈ On → (A ·o ∅) ∈ On)
13		omsuc 3133	. . . . . . . . . 10 ⊢ ((A ∈ On ∧ y ∈ On) → (A ·o suc y) = ((A ·o y) +o A))
14	13	eleq1d 1155	. . . . . . . . 9 ⊢ ((A ∈ On ∧ y ∈ On) → ((A ·o suc y) ∈ On ↔ ((A ·o y) +o A) ∈ On))
15		oacl 3138	. . . . . . . . 9 ⊢ (((A ·o y) ∈ On ∧ A ∈ On) → ((A ·o y) +o A) ∈ On)
16	14, 15	syl5bir 184	. . . . . . . 8 ⊢ ((A ∈ On ∧ y ∈ On) → (((A ·o y) ∈ On ∧ A ∈ On) → (A ·o suc y) ∈ On))
17	16	exp4b 296	. . . . . . 7 ⊢ (A ∈ On → (y ∈ On → ((A ·o y) ∈ On → (A ∈ On → (A ·o suc y) ∈ On))))
18	17	com24 37	. . . . . 6 ⊢ (A ∈ On → (A ∈ On → ((A ·o y) ∈ On → (y ∈ On → (A ·o suc y) ∈ On))))
19	18	pm2.43i 58	. . . . 5 ⊢ (A ∈ On → ((A ·o y) ∈ On → (y ∈ On → (A ·o suc y) ∈ On)))
20	19	com3r 35	. . . 4 ⊢ (y ∈ On → (A ∈ On → ((A ·o y) ∈ On → (A ·o suc y) ∈ On)))
21		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
22		omlim 3136	. . . . . . . . 9 ⊢ ((A ∈ On ∧ (x ∈ V ∧ Lim x)) → (A ·o x) = ∪y ∈ x (A ·o y))
23	21, 22	mpan21 531	. . . . . . . 8 ⊢ ((A ∈ On ∧ Lim x) → (A ·o x) = ∪y ∈ x (A ·o y))
24	23	ancoms 334	. . . . . . 7 ⊢ ((Lim x ∧ A ∈ On) → (A ·o x) = ∪y ∈ x (A ·o y))
25	24	eleq1d 1155	. . . . . 6 ⊢ ((Lim x ∧ A ∈ On) → ((A ·o x) ∈ On ↔ ∪y ∈ x (A ·o y) ∈ On))
26		oprex 3018	. . . . . . 7 ⊢ (A ·o y) ∈ V
27	21, 26	iunon 2947	. . . . . 6 ⊢ (∀y ∈ x (A ·o y) ∈ On → ∪y ∈ x (A ·o y) ∈ On)
28	25, 27	syl5bir 184	. . . . 5 ⊢ ((Lim x ∧ A ∈ On) → (∀y ∈ x (A ·o y) ∈ On → (A ·o x) ∈ On))
29	28	exp 291	. . . 4 ⊢ (Lim x → (A ∈ On → (∀y ∈ x (A ·o y) ∈ On → (A ·o x) ∈ On)))
30	2, 4, 6, 8, 12, 20, 29	tfinds3 2406	. . 3 ⊢ (B ∈ On → (A ∈ On → (A ·o B) ∈ On))
31	30	com12 13	. 2 ⊢ (A ∈ On → (B ∈ On → (A ·o B) ∈ On))
32	31	imp 277	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ·o B) ∈ On)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∅c0 1707  ∪ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  oecl 3140  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-oadd 3106  df-omul 3107

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