Theorem oecl 5218

Description: Closure law for ordinal exponentiation. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

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 4889	. . . . 5 |- (A = (/) -> (A ^o B) = ((/) ^o B))
2	1	eleq1d 1963	. . . 4 |- (A = (/) -> ((A ^o B) e. On <-> ((/) ^o B) e. On))
3		opreq2 4890	. . . . . . . 8 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
4		oe0m0 5204	. . . . . . . . 9 |- ((/) ^o (/)) = 1o
5		1on 5182	. . . . . . . . 9 |- 1o e. On
6	4, 5	eqeltri 1967	. . . . . . . 8 |- ((/) ^o (/)) e. On
7	3, 6	syl6eqel 1979	. . . . . . 7 |- (B = (/) -> ((/) ^o B) e. On)
8	7	adantl 424	. . . . . 6 |- ((B e. On /\ B = (/)) -> ((/) ^o B) e. On)
9		oe0m1 5205	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
10	9	biimpa 460	. . . . . . . 8 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
11		0elon 3716	. . . . . . . 8 |- (/) e. On
12	10, 11	syl6eqel 1979	. . . . . . 7 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) e. On)
13	12	adantll 428	. . . . . 6 |- (((B e. On /\ B e. On) /\ (/) e. B) -> ((/) ^o B) e. On)
14	8, 13	oe0lem 5197	. . . . 5 |- ((B e. On /\ B e. On) -> ((/) ^o B) e. On)
15	14	anidms 480	. . . 4 |- (B e. On -> ((/) ^o B) e. On)
16	2, 15	syl5bir 227	. . 3 |- (A = (/) -> (B e. On -> (A ^o B) e. On))
17	16	impcom 378	. 2 |- ((B e. On /\ A = (/)) -> (A ^o B) e. On)
18		opreq2 4890	. . . . . . 7 |- (x = (/) -> (A ^o x) = (A ^o (/)))
19	18	eleq1d 1963	. . . . . 6 |- (x = (/) -> ((A ^o x) e. On <-> (A ^o (/)) e. On))
20		opreq2 4890	. . . . . . 7 |- (x = y -> (A ^o x) = (A ^o y))
21	20	eleq1d 1963	. . . . . 6 |- (x = y -> ((A ^o x) e. On <-> (A ^o y) e. On))
22		opreq2 4890	. . . . . . 7 |- (x = suc y -> (A ^o x) = (A ^o suc y))
23	22	eleq1d 1963	. . . . . 6 |- (x = suc y -> ((A ^o x) e. On <-> (A ^o suc y) e. On))
24		opreq2 4890	. . . . . . 7 |- (x = B -> (A ^o x) = (A ^o B))
25	24	eleq1d 1963	. . . . . 6 |- (x = B -> ((A ^o x) e. On <-> (A ^o B) e. On))
26		oe0 5206	. . . . . . . 8 |- (A e. On -> (A ^o (/)) = 1o)
27	26, 5	syl6eqel 1979	. . . . . . 7 |- (A e. On -> (A ^o (/)) e. On)
28	27	adantr 425	. . . . . 6 |- ((A e. On /\ (/) e. A) -> (A ^o (/)) e. On)
29		omcl 5216	. . . . . . . . . . 11 |- (((A ^o y) e. On /\ A e. On) -> ((A ^o y) .o A) e. On)
30	29	expcom 403	. . . . . . . . . 10 |- (A e. On -> ((A ^o y) e. On -> ((A ^o y) .o A) e. On))
31	30	adantr 425	. . . . . . . . 9 |- ((A e. On /\ y e. On) -> ((A ^o y) e. On -> ((A ^o y) .o A) e. On))
32		oesuc 5211	. . . . . . . . . 10 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
33	32	eleq1d 1963	. . . . . . . . 9 |- ((A e. On /\ y e. On) -> ((A ^o suc y) e. On <-> ((A ^o y) .o A) e. On))
34	31, 33	sylibrd 221	. . . . . . . 8 |- ((A e. On /\ y e. On) -> ((A ^o y) e. On -> (A ^o suc y) e. On))
35	34	expcom 403	. . . . . . 7 |- (y e. On -> (A e. On -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
36	35	adantrd 427	. . . . . 6 |- (y e. On -> ((A e. On /\ (/) e. A) -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
37		visset 2295	. . . . . . . . . . . 12 |- x e. _V
38		oelim 5214	. . . . . . . . . . . 12 |- (((A e. On /\ (x e. _V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
39	37, 38	mpanlr1 779	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
40	39	anasss 488	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
41	40	an1s 544	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
42	41	eleq1d 1963	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((A ^o x) e. On <-> U_y e. x (A ^o y) e. On))
43		oprex 4907	. . . . . . . . 9 |- (A ^o y) e. _V
44	37, 43	iunon 5114	. . . . . . . 8 |- (A.y e. x (A ^o y) e. On -> U_y e. x (A ^o y) e. On)
45	42, 44	syl5bir 227	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On))
46	45	ex 402	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On)))
47	19, 21, 23, 25, 28, 36, 46	tfinds3 3948	. . . . 5 |- (B e. On -> ((A e. On /\ (/) e. A) -> (A ^o B) e. On))
48	47	exp3a 405	. . . 4 |- (B e. On -> (A e. On -> ((/) e. A -> (A ^o B) e. On)))
49	48	com12 14	. . 3 |- (A e. On -> (B e. On -> ((/) e. A -> (A ^o B) e. On)))
50	49	imp31 389	. 2 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) e. On)
51	17, 50	oe0lem 5197	1 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  (/)c0 2875  U_ciun 3255  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884  1oc1o 5172   .o comu 5175   ^o coe 5176

This theorem is referenced by:  oen0 5261  oeordi 5262  oeord 5263  oecan 5264  oeword 5265  oewordri 5267  oeworde 5268  oeordsuc 5269  oeoalem 5271  oeoa 5272  oeoelem 5273  oeoe 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-oexp 5181

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