Theorem oecl 4172

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 3968	. . . . 5 |- (A = (/) -> (A ^o B) = ((/) ^o B))
2	1	eleq1d 1540	. . . 4 |- (A = (/) -> ((A ^o B) e. On <-> ((/) ^o B) e. On))
3		opreq2 3969	. . . . . . . 8 |- (B = (/) -> ((/) ^o B) = ((/) ^o (/)))
4		oe0m0 4159	. . . . . . . . 9 |- ((/) ^o (/)) = 1o
5		1on 4138	. . . . . . . . 9 |- 1o e. On
6	4, 5	eqeltr 1544	. . . . . . . 8 |- ((/) ^o (/)) e. On
7	3, 6	syl6eqel 1556	. . . . . . 7 |- (B = (/) -> ((/) ^o B) e. On)
8	7	adantl 388	. . . . . 6 |- ((B e. On /\ B = (/)) -> ((/) ^o B) e. On)
9		oe0m1 4160	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
10	9	biimpa 416	. . . . . . . 8 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
11		0elon 3022	. . . . . . . 8 |- (/) e. On
12	10, 11	syl6eqel 1556	. . . . . . 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 4152	. . . . 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 3969	. . . . . . 7 |- (x = (/) -> (A ^o x) = (A ^o (/)))
19	18	eleq1d 1540	. . . . . 6 |- (x = (/) -> ((A ^o x) e. On <-> (A ^o (/)) e. On))
20		opreq2 3969	. . . . . . 7 |- (x = y -> (A ^o x) = (A ^o y))
21	20	eleq1d 1540	. . . . . 6 |- (x = y -> ((A ^o x) e. On <-> (A ^o y) e. On))
22		opreq2 3969	. . . . . . 7 |- (x = suc y -> (A ^o x) = (A ^o suc y))
23	22	eleq1d 1540	. . . . . 6 |- (x = suc y -> ((A ^o x) e. On <-> (A ^o suc y) e. On))
24		opreq2 3969	. . . . . . 7 |- (x = B -> (A ^o x) = (A ^o B))
25	24	eleq1d 1540	. . . . . 6 |- (x = B -> ((A ^o x) e. On <-> (A ^o B) e. On))
26		oe0 4161	. . . . . . . 8 |- (A e. On -> (A ^o (/)) = 1o)
27	26, 5	syl6eqel 1556	. . . . . . 7 |- (A e. On -> (A ^o (/)) e. On)
28	27	adantr 389	. . . . . 6 |- ((A e. On /\ (/) e. A) -> (A ^o (/)) e. On)
29		oesuc 4166	. . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
30	29	eleq1d 1540	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> ((A ^o suc y) e. On <-> ((A ^o y) .o A) e. On))
31		omcl 4171	. . . . . . . . . . . 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 1813	. . . . . . . . . . . 12 |- x e. V
39		oelim 4169	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 486	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
43	42	eleq1d 1540	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((A ^o x) e. On <-> U_y e. x (A ^o y) e. On))
44		oprex 3983	. . . . . . . . 9 |- (A ^o y) e. V
45	38, 44	iunon 3909	. . . . . . . 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 3166	. . . . 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 4152	1 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  (/)c0 2280  U_ciun 2566  Oncon0 2948  Lim wlim 2949  suc csuc 2950  (class class class)co 3963  1oc1o 4128   .o comu 4131   ^o coe 4132

This theorem is referenced by:  oen0 4213  oeordi 4214  oeord 4215  oecan 4216  oeword 4217  oewordri 4219  oeworde 4220  oeordsuc 4221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-omul 4136  df-oexp 4137

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