Theorem oalimcl 5242

Description: The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60.

Assertion

Ref	Expression
oalimcl	|- ((A e. On /\ (B e. C /\ Lim B)) -> Lim (A +o B))

Proof of Theorem oalimcl

Step	Hyp	Ref	Expression
1		dflim3 3930	. 2 |- (Lim (A +o B) <-> (Ord (A +o B) /\ -. ((A +o B) = (/) \/ E.y e. On (A +o B) = suc y)))
2		oacl 5215	. . . 4 |- ((A e. On /\ B e. On) -> (A +o B) e. On)
3		eloni 3667	. . . 4 |- ((A +o B) e. On -> Ord (A +o B))
4	2, 3	syl 12	. . 3 |- ((A e. On /\ B e. On) -> Ord (A +o B))
5		limelon 3727	. . 3 |- ((B e. C /\ Lim B) -> B e. On)
6	4, 5	sylan2 500	. 2 |- ((A e. On /\ (B e. C /\ Lim B)) -> Ord (A +o B))
7		ioran 331	. . 3 |- (-. ((A +o B) = (/) \/ E.y e. On (A +o B) = suc y) <-> (-. (A +o B) = (/) /\ -. E.y e. On (A +o B) = suc y))
8		0ellim 3726	. . . . . 6 |- (Lim B -> (/) e. B)
9		n0i 2880	. . . . . 6 |- ((/) e. B -> -. B = (/))
10	8, 9	syl 12	. . . . 5 |- (Lim B -> -. B = (/))
11	10	ad2antll 443	. . . 4 |- ((A e. On /\ (B e. C /\ Lim B)) -> -. B = (/))
12		oa00 5241	. . . . . . 7 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) <-> (A = (/) /\ B = (/))))
13		simpr 350	. . . . . . 7 |- ((A = (/) /\ B = (/)) -> B = (/))
14	12, 13	syl6bi 231	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) -> B = (/)))
15	14	con3d 111	. . . . 5 |- ((A e. On /\ B e. On) -> (-. B = (/) -> -. (A +o B) = (/)))
16	15, 5	sylan2 500	. . . 4 |- ((A e. On /\ (B e. C /\ Lim B)) -> (-. B = (/) -> -. (A +o B) = (/)))
17	11, 16	mpd 29	. . 3 |- ((A e. On /\ (B e. C /\ Lim B)) -> -. (A +o B) = (/))
18		eqeq1 1890	. . . . . . . . . . . 12 |- ((A +o B) = suc y -> ((A +o B) = U_x e. B (A +o x) <-> suc y = U_x e. B (A +o x)))
19		oalim 5212	. . . . . . . . . . . 12 |- ((A e. On /\ (B e. C /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))
20	18, 19	syl5bi 225	. . . . . . . . . . 11 |- ((A +o B) = suc y -> ((A e. On /\ (B e. C /\ Lim B)) -> suc y = U_x e. B (A +o x)))
21	20	imp 377	. . . . . . . . . 10 |- (((A +o B) = suc y /\ (A e. On /\ (B e. C /\ Lim B))) -> suc y = U_x e. B (A +o x))
22		visset 2295	. . . . . . . . . . 11 |- y e. _V
23	22	sucid 3744	. . . . . . . . . 10 |- y e. suc y
24	21, 23	syl5eleq 1977	. . . . . . . . 9 |- (((A +o B) = suc y /\ (A e. On /\ (B e. C /\ Lim B))) -> y e. U_x e. B (A +o x))
25		eliun 3259	. . . . . . . . 9 |- (y e. U_x e. B (A +o x) <-> E.x e. B y e. (A +o x))
26	24, 25	sylib 215	. . . . . . . 8 |- (((A +o B) = suc y /\ (A e. On /\ (B e. C /\ Lim B))) -> E.x e. B y e. (A +o x))
27		onelon 3683	. . . . . . . . . . . . . . . 16 |- ((B e. On /\ x e. B) -> x e. On)
28	27, 5	sylan 497	. . . . . . . . . . . . . . 15 |- (((B e. C /\ Lim B) /\ x e. B) -> x e. On)
29		onnbtwn 3762	. . . . . . . . . . . . . . . . . 18 |- (x e. On -> -. (x e. B /\ B e. suc x))
30		imnan 261	. . . . . . . . . . . . . . . . . 18 |- ((x e. B -> -. B e. suc x) <-> -. (x e. B /\ B e. suc x))
31	29, 30	sylibr 217	. . . . . . . . . . . . . . . . 17 |- (x e. On -> (x e. B -> -. B e. suc x))
32	31	com12 14	. . . . . . . . . . . . . . . 16 |- (x e. B -> (x e. On -> -. B e. suc x))
33	32	adantl 424	. . . . . . . . . . . . . . 15 |- (((B e. C /\ Lim B) /\ x e. B) -> (x e. On -> -. B e. suc x))
34	28, 33	mpd 29	. . . . . . . . . . . . . 14 |- (((B e. C /\ Lim B) /\ x e. B) -> -. B e. suc x)
35	34	ad2antrl 442	. . . . . . . . . . . . 13 |- ((A e. On /\ (((B e. C /\ Lim B) /\ x e. B) /\ y e. (A +o x))) -> -. B e. suc x)
36		oacl 5215	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A e. On /\ x e. On) -> (A +o x) e. On)
37		eloni 3667	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A +o x) e. On -> Ord (A +o x))
38		ordsucelsuc 3902	. . . . . . . . . . . . . . . . . . . . . 22 |- (Ord (A +o x) -> (y e. (A +o x) <-> suc y e. suc (A +o x)))
39	36, 37, 38	3syl 24	. . . . . . . . . . . . . . . . . . . . 21 |- ((A e. On /\ x e. On) -> (y e. (A +o x) <-> suc y e. suc (A +o x)))
40		oasuc 5208	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A e. On /\ x e. On) -> (A +o suc x) = suc (A +o x))
41	40	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . 21 |- ((A e. On /\ x e. On) -> (suc y e. (A +o suc x) <-> suc y e. suc (A +o x)))
42	39, 41	bitr4d 590	. . . . . . . . . . . . . . . . . . . 20 |- ((A e. On /\ x e. On) -> (y e. (A +o x) <-> suc y e. (A +o suc x)))
43	42, 28	sylan2 500	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ ((B e. C /\ Lim B) /\ x e. B)) -> (y e. (A +o x) <-> suc y e. (A +o suc x)))
44		eleq1 1957	. . . . . . . . . . . . . . . . . . . 20 |- ((A +o B) = suc y -> ((A +o B) e. (A +o suc x) <-> suc y e. (A +o suc x)))
45	44	bicomd 580	. . . . . . . . . . . . . . . . . . 19 |- ((A +o B) = suc y -> (suc y e. (A +o suc x) <-> (A +o B) e. (A +o suc x)))
46	43, 45	sylan9bbr 600	. . . . . . . . . . . . . . . . . 18 |- (((A +o B) = suc y /\ (A e. On /\ ((B e. C /\ Lim B) /\ x e. B))) -> (y e. (A +o x) <-> (A +o B) e. (A +o suc x)))
47		oaord 5228	. . . . . . . . . . . . . . . . . . . . . 22 |- ((B e. On /\ suc x e. On /\ A e. On) -> (B e. suc x <-> (A +o B) e. (A +o suc x)))
48	47	3expa 1067	. . . . . . . . . . . . . . . . . . . . 21 |- (((B e. On /\ suc x e. On) /\ A e. On) -> (B e. suc x <-> (A +o B) e. (A +o suc x)))
49	5	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. C /\ Lim B) /\ x e. B) -> B e. On)
50		sucelon 3898	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. On <-> suc x e. On)
51	28, 50	sylib 215	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. C /\ Lim B) /\ x e. B) -> suc x e. On)
52	49, 51	jca 310	. . . . . . . . . . . . . . . . . . . . 21 |- (((B e. C /\ Lim B) /\ x e. B) -> (B e. On /\ suc x e. On))
53	48, 52	sylan 497	. . . . . . . . . . . . . . . . . . . 20 |- ((((B e. C /\ Lim B) /\ x e. B) /\ A e. On) -> (B e. suc x <-> (A +o B) e. (A +o suc x)))
54	53	ancoms 484	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ ((B e. C /\ Lim B) /\ x e. B)) -> (B e. suc x <-> (A +o B) e. (A +o suc x)))
55	54	adantl 424	. . . . . . . . . . . . . . . . . 18 |- (((A +o B) = suc y /\ (A e. On /\ ((B e. C /\ Lim B) /\ x e. B))) -> (B e. suc x <-> (A +o B) e. (A +o suc x)))
56	46, 55	bitr4d 590	. . . . . . . . . . . . . . . . 17 |- (((A +o B) = suc y /\ (A e. On /\ ((B e. C /\ Lim B) /\ x e. B))) -> (y e. (A +o x) <-> B e. suc x))
57	56	biimpd 170	. . . . . . . . . . . . . . . 16 |- (((A +o B) = suc y /\ (A e. On /\ ((B e. C /\ Lim B) /\ x e. B))) -> (y e. (A +o x) -> B e. suc x))
58	57	exp32 408	. . . . . . . . . . . . . . 15 |- ((A +o B) = suc y -> (A e. On -> (((B e. C /\ Lim B) /\ x e. B) -> (y e. (A +o x) -> B e. suc x))))
59	58	com4l 43	. . . . . . . . . . . . . 14 |- (A e. On -> (((B e. C /\ Lim B) /\ x e. B) -> (y e. (A +o x) -> ((A +o B) = suc y -> B e. suc x))))
60	59	imp32 390	. . . . . . . . . . . . 13 |- ((A e. On /\ (((B e. C /\ Lim B) /\ x e. B) /\ y e. (A +o x))) -> ((A +o B) = suc y -> B e. suc x))
61	35, 60	mtod 123	. . . . . . . . . . . 12 |- ((A e. On /\ (((B e. C /\ Lim B) /\ x e. B) /\ y e. (A +o x))) -> -. (A +o B) = suc y)
62	61	exp44 416	. . . . . . . . . . 11 |- (A e. On -> ((B e. C /\ Lim B) -> (x e. B -> (y e. (A +o x) -> -. (A +o B) = suc y))))
63	62	imp 377	. . . . . . . . . 10 |- ((A e. On /\ (B e. C /\ Lim B)) -> (x e. B -> (y e. (A +o x) -> -. (A +o B) = suc y)))
64	63	r19.23adv 2215	. . . . . . . . 9 |- ((A e. On /\ (B e. C /\ Lim B)) -> (E.x e. B y e. (A +o x) -> -. (A +o B) = suc y))
65	64	adantl 424	. . . . . . . 8 |- (((A +o B) = suc y /\ (A e. On /\ (B e. C /\ Lim B))) -> (E.x e. B y e. (A +o x) -> -. (A +o B) = suc y))
66	26, 65	mpd 29	. . . . . . 7 |- (((A +o B) = suc y /\ (A e. On /\ (B e. C /\ Lim B))) -> -. (A +o B) = suc y)
67	66	expcom 403	. . . . . 6 |- ((A e. On /\ (B e. C /\ Lim B)) -> ((A +o B) = suc y -> -. (A +o B) = suc y))
68	67	pm2.01d 105	. . . . 5 |- ((A e. On /\ (B e. C /\ Lim B)) -> -. (A +o B) = suc y)
69	68	adantr 425	. . . 4 |- (((A e. On /\ (B e. C /\ Lim B)) /\ y e. On) -> -. (A +o B) = suc y)
70	69	nrexdv 2193	. . 3 |- ((A e. On /\ (B e. C /\ Lim B)) -> -. E.y e. On (A +o B) = suc y)
71	7, 17, 70	sylanbrc 527	. 2 |- ((A e. On /\ (B e. C /\ Lim B)) -> -. ((A +o B) = (/) \/ E.y e. On (A +o B) = suc y))
72	1, 6, 71	sylanbrc 527	1 |- ((A e. On /\ (B e. C /\ Lim B)) -> Lim (A +o B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  (/)c0 2875  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  oaass 5243  odi 5258

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-oadd 5179

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