Theorem oecl 4156

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 3953	. . . . 5 |- (A = (/) -> (A ^o B) = ((/) ^o B))
2	1	eleq1d 1532	. . . 4 |- (A = (/) -> ((A ^o B) e. On <-> ((/) ^o B) e. On))
3		opreq2 3954	. . . . . . . 8 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
4		oe0m0 4143	. . . . . . . . 9 |- ((/) ^o (/)) = 1o
5		1on 4122	. . . . . . . . 9 |- 1o e. On
6	4, 5	eqeltr 1536	. . . . . . . 8 |- ((/) ^o (/)) e. On
7	3, 6	syl6eqel 1548	. . . . . . 7 |- (B = (/) -> ((/) ^o B) e. On)
8	7	adantl 388	. . . . . 6 |- ((B e. On /\ B = (/)) -> ((/) ^o B) e. On)
9		oe0m1 4144	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
10	9	biimpa 416	. . . . . . . 8 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
11		0elon 3012	. . . . . . . 8 |- (/) e. On
12	10, 11	syl6eqel 1548	. . . . . . 7 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) e. On)
13	12	adantll 392	. . . . . 6 |- (((B e. On /\ B e. On) /\ (/) e. B) -> ((/) ^o B) e. On)
14	8, 13	oe0lem 4136	. . . . 5 |- ((B e. On /\ B e. On) -> ((/) ^o B) e. On)
15	14	anidms 434	. . . 4 |- (B e. On -> ((/) ^o B) e. On)
16	2, 15	syl5bir 210	. . 3 |- (A = (/) -> (B e. On -> (A ^o B) e. On))
17	16	impcom 351	. 2 |- ((B e. On /\ A = (/)) -> (A ^o B) e. On)
18		opreq2 3954	. . . . . . 7 |- (x = (/) -> (A ^o x) = (A ^o (/)))
19	18	eleq1d 1532	. . . . . 6 |- (x = (/) -> ((A ^o x) e. On <-> (A ^o (/)) e. On))
20		opreq2 3954	. . . . . . 7 |- (x = y -> (A ^o x) = (A ^o y))
21	20	eleq1d 1532	. . . . . 6 |- (x = y -> ((A ^o x) e. On <-> (A ^o y) e. On))
22		opreq2 3954	. . . . . . 7 |- (x = suc y -> (A ^o x) = (A ^o suc y))
23	22	eleq1d 1532	. . . . . 6 |- (x = suc y -> ((A ^o x) e. On <-> (A ^o suc y) e. On))
24		opreq2 3954	. . . . . . 7 |- (x = B -> (A ^o x) = (A ^o B))
25	24	eleq1d 1532	. . . . . 6 |- (x = B -> ((A ^o x) e. On <-> (A ^o B) e. On))
26		oe0 4145	. . . . . . . 8 |- (A e. On -> (A ^o (/)) = 1o)
27	26, 5	syl6eqel 1548	. . . . . . 7 |- (A e. On -> (A ^o (/)) e. On)
28	27	adantr 389	. . . . . 6 |- ((A e. On /\ (/) e. A) -> (A ^o (/)) e. On)
29		oesuc 4150	. . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
30	29	eleq1d 1532	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> ((A ^o suc y) e. On <-> ((A ^o y) .o A) e. On))
31		omcl 4155	. . . . . . . . . . . 12 |- (((A ^o y) e. On /\ A e. On) -> ((A ^o y) .o A) e. On)
32	30, 31	syl5bir 210	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (((A ^o y) e. On /\ A e. On) -> (A ^o suc y) e. On))
33	32	exp4b 379	. . . . . . . . . 10 |- (A e. On -> (y e. On -> ((A ^o y) e. On -> (A e. On -> (A ^o suc y) e. On))))
34	33	com24 37	. . . . . . . . 9 |- (A e. On -> (A e. On -> ((A ^o y) e. On -> (y e. On -> (A ^o suc y) e. On))))
35	34	pm2.43i 64	. . . . . . . 8 |- (A e. On -> ((A ^o y) e. On -> (y e. On -> (A ^o suc y) e. On)))
36	35	com3r 35	. . . . . . 7 |- (y e. On -> (A e. On -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
37	36	adantrd 391	. . . . . 6 |- (y e. On -> ((A e. On /\ (/) e. A) -> ((A ^o y) e. On -> (A ^o suc y) e. On)))
38		visset 1804	. . . . . . . . . . . 12 |- x e. V
39		oelim 4153	. . . . . . . . . . . 12 |- (((A e. On /\ (x e. V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
40	38, 39	mpanlr1 709	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
41	40	anasss 440	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
42	41	an1s 485	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
43	42	eleq1d 1532	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((A ^o x) e. On <-> U_y e. x (A ^o y) e. On))
44		oprex 3968	. . . . . . . . 9 |- (A ^o y) e. V
45	38, 44	iunon 3894	. . . . . . . 8 |- (A.y e. x (A ^o y) e. On -> U_y e. x (A ^o y) e. On)
46	43, 45	syl5bir 210	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On))
47	46	ex 373	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x (A ^o y) e. On -> (A ^o x) e. On)))
48	19, 21, 23, 25, 28, 37, 47	tfinds3 3156	. . . . 5 |- (B e. On -> ((A e. On /\ (/) e. A) -> (A ^o B) e. On))
49	48	exp3a 375	. . . 4 |- (B e. On -> (A e. On -> ((/) e. A -> (A ^o B) e. On)))
50	49	com12 11	. . 3 |- (A e. On -> (B e. On -> ((/) e. A -> (A ^o B) e. On)))
51	50	imp31 362	. 2 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) e. On)
52	17, 51	oe0lem 4136	1 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802  (/)c0 2270  U_ciun 2556  Oncon0 2938  Lim wlim 2939  suc csuc 2940  (class class class)co 3948  1oc1o 4112   .o comu 4115   ^o coe 4116

This theorem is referenced by:  oen0 4197  oeordi 4198  oeord 4199  oecan 4200  oeword 4201  oewordri 4203  oeworde 4204  oeordsuc 4205

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-oadd 4119  df-omul 4120  df-oexp 4121

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