Theorem oecl 3140

Description: Closure law for ordinal exponentiation.

Assertion

Ref	Expression
oecl	|- ((A e. On /\ B e. On) -> (A ^o B) e. On)

Proof of Theorem oecl

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . . 6 |- (A = (/) -> (A ^o B) = ((/) ^o B))
2	1	eleq1d 1155	. . . . 5 |- (A = (/) -> ((A ^o B) e. On <-> ((/) ^o B) e. On))
3		oe0m0 3128	. . . . . . . . . 10 |- ((/) ^o (/)) = 1o
4		1o 3109	. . . . . . . . . 10 |- 1o e. On
5	3, 4	eqeltr 1159	. . . . . . . . 9 |- ((/) ^o (/)) e. On
6		opreq2 3007	. . . . . . . . . 10 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
7	6	eleq1d 1155	. . . . . . . . 9 |- (B = (/) -> (((/) ^o B) e. On <-> ((/) ^o (/)) e. On))
8	5, 7	mpbiri 169	. . . . . . . 8 |- (B = (/) -> ((/) ^o B) e. On)
9	8	adantl 305	. . . . . . 7 |- ((B e. On /\ B = (/)) -> ((/) ^o B) e. On)
10		0elon 2277	. . . . . . . . 9 |- (/) e. On
11		oe0m1 3129	. . . . . . . . . 10 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
12	11	eleq1d 1155	. . . . . . . . 9 |- ((B e. On /\ (/) e. B) -> (((/) ^o B) e. On <-> (/) e. On))
13	10, 12	mpbiri 169	. . . . . . . 8 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) e. On)
14	13	adantll 309	. . . . . . 7 |- (((B e. On /\ B e. On) /\ (/) e. B) -> ((/) ^o B) e. On)
15	9, 14	oe0lem 3121	. . . . . 6 |- ((B e. On /\ B e. On) -> ((/) ^o B) e. On)
16	15	anidms 332	. . . . 5 |- (B e. On -> ((/) ^o B) e. On)
17	2, 16	syl5bir 184	. . . 4 |- (A = (/) -> (B e. On -> (A ^o B) e. On))
18	17	com12 13	. . 3 |- (B e. On -> (A = (/) -> (A ^o B) e. On))
19	18	imp 277	. 2 |- ((B e. On /\ A = (/)) -> (A ^o B) e. On)
20		opreq2 3007	. . . . . . 7 |- (x = (/) -> (A ^o x) = (A ^o (/)))
21	20	eleq1d 1155	. . . . . 6 |- (x = (/) -> ((A ^o x) e. On <-> (A ^o (/)) e. On))
22		opreq2 3007	. . . . . . 7 |- (x = y -> (A ^o x) = (A ^o y))
23	22	eleq1d 1155	. . . . . 6 |- (x = y -> ((A ^o x) e. On <-> (A ^o y) e. On))
24		opreq2 3007	. . . . . . 7 |- (x = suc y -> (A ^o x) = (A ^o suc y))
25	24	eleq1d 1155	. . . . . 6 |- (x = suc y -> ((A ^o x) e. On <-> (A ^o suc y) e. On))
26		opreq2 3007	. . . . . . 7 |- (x = B -> (A ^o x) = (A ^o B))
27	26	eleq1d 1155	. . . . . 6 |- (x = B -> ((A ^o x) e. On <-> (A ^o B) e. On))
28		oe0 3130	. . . . . . . . 9 |- (A e. On -> (A ^o (/)) = 1o)
29	28	eleq1d 1155	. . . . . . . 8 |- (A e. On -> ((A ^o (/)) e. On <-> 1o e. On))
30	4, 29	mpbiri 169	. . . . . . 7 |- (A e. On -> (A ^o (/)) e. On)
31	30	adantr 306	. . . . . 6 |- ((A e. On /\ (/) e. A) -> (A ^o (/)) e. On)
32		oesuc 3134	. . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
33	32	eleq1d 1155	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> ((A ^o suc y) e. On <-> ((A ^o y) .o A) e. On))
34		omcl 3139	. . . . . . . . . . . 12 |- (((A ^o y) e. On /\ A e. On) -> ((A ^o y) .o A) e. On)
35	33, 34	syl5bir 184	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (((A ^o y) e. On /\ A e. On) -> (A ^o suc y) e. On))
36	35	exp4b 296	. . . . . . . . . 10 |- (A e. On -> (y e. On -> ((A ^o y) e. On -> (A e. On -> (A ^o suc y) e. On))))
37	36	com24 37	. . . . . . . . 9 |- (A e. On -> (A e. On -> ((A ^o y) e. On -> (y e. On -> (A ^o suc y) e. On))))
38	37	pm2.43i 58	. . . . . . . 8 |- (A e. On -> ((A ^o y) e. On -> (y e. On -> (A ^o suc y) e. On)))
39	38	com3r 35	. . . . . . 7 |- (y e. On -> (A e. On -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
40	39	adantrd 308	. . . . . 6 |- (y e. On -> ((A e. On /\ (/) e. A) -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
41		visset 1350	. . . . . . . . . . . 12 |- x e. V
42		oelim 3137	. . . . . . . . . . . 12 |- (((A e. On /\ (x e. V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U.y e. x (A ^o y))
43	41, 42	mpan121 533	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U.y e. x (A ^o y))
44	43	anasss 337	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U.y e. x (A ^o y))
45	44	an1s 372	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U.y e. x (A ^o y))
46	45	eleq1d 1155	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((A ^o x) e. On <-> U.y e. x (A ^o y) e. On))
47		oprex 3018	. . . . . . . . 9 |- (A ^o y) e. V
48	41, 47	iunon 2947	. . . . . . . 8 |- (A.y e. x (A ^o y) e. On -> U.y e. x (A ^o y) e. On)
49	46, 48	syl5bir 184	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On))
50	49	exp 291	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On)))
51	21, 23, 25, 27, 31, 40, 50	tfinds3 2406	. . . . 5 |- (B e. On -> ((A e. On /\ (/) e. A) -> (A ^o B) e. On))
52	51	exp3a 292	. . . 4 |- (B e. On -> (A e. On -> ((/) e. A -> (A ^o B) e. On)))
53	52	com12 13	. . 3 |- (A e. On -> (B e. On -> ((/) e. A -> (A ^o B) e. On)))
54	53	imp31 280	. 2 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) e. On)
55	19, 54	oe0lem 3121	1 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)

Syntax hints:   -> wi 2   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348  (/)c0 1707  U.ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001  1oc1o 3099   .o comu 3102   ^o coe 3103

This theorem is referenced by:  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-oadd 3106  df-omul 3107  df-oexp 3108

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