Theorem tolat 14631

Description: A totally ordered set is a lattice.

Assertion

Ref	Expression
tolat	|- Toset C_ Lat

Proof of Theorem tolat

Step	Hyp	Ref	Expression
1		dfss2 2610	. 2 |- ( Toset C_ Lat <-> A.x(x e. Toset -> x e. Lat))
2		visset 2295	. . . . . . . . . . . . . . 15 |- v e. _V
3		spwpr4c 10009	. . . . . . . . . . . . . . . 16 |- ((x e. Poset /\ v e. _V /\ uxv) -> (x supw {u, v}) = v)
4	3	3exp 1066	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (v e. _V -> (uxv -> (x supw {u, v}) = v)))
5	2, 4	mpi 55	. . . . . . . . . . . . . 14 |- (x e. Poset -> (uxv -> (x supw {u, v}) = v))
6	5	impcom 378	. . . . . . . . . . . . 13 |- ((uxv /\ x e. Poset) -> (x supw {u, v}) = v)
7		posispre 14582	. . . . . . . . . . . . . . . 16 |- (x e. Poset -> x e. Preset )
8		eqid 1884	. . . . . . . . . . . . . . . . . 18 |- dom x = dom x
9	8	pre2befi2 14573	. . . . . . . . . . . . . . . . 17 |- ((x e. Preset /\ v e. _V /\ uxv) -> v e. dom x)
10	9	3exp 1066	. . . . . . . . . . . . . . . 16 |- (x e. Preset -> (v e. _V -> (uxv -> v e. dom x)))
11	7, 10	syl 12	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (v e. _V -> (uxv -> v e. dom x)))
12	2, 11	mpi 55	. . . . . . . . . . . . . 14 |- (x e. Poset -> (uxv -> v e. dom x))
13	12	impcom 378	. . . . . . . . . . . . 13 |- ((uxv /\ x e. Poset) -> v e. dom x)
14	6, 13	eqeltrd 1971	. . . . . . . . . . . 12 |- ((uxv /\ x e. Poset) -> (x supw {u, v}) e. dom x)
15		nfwpr4c 14630	. . . . . . . . . . . . . . . 16 |- ((x e. Poset /\ v e. _V /\ uxv) -> (x infw {u, v}) = u)
16	15	3exp 1066	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (v e. _V -> (uxv -> (x infw {u, v}) = u)))
17	2, 16	mpi 55	. . . . . . . . . . . . . 14 |- (x e. Poset -> (uxv -> (x infw {u, v}) = u))
18	17	impcom 378	. . . . . . . . . . . . 13 |- ((uxv /\ x e. Poset) -> (x infw {u, v}) = u)
19		visset 2295	. . . . . . . . . . . . . . 15 |- u e. _V
20	19	breldm 4161	. . . . . . . . . . . . . 14 |- (uxv -> u e. dom x)
21	20	adantr 425	. . . . . . . . . . . . 13 |- ((uxv /\ x e. Poset) -> u e. dom x)
22	18, 21	eqeltrd 1971	. . . . . . . . . . . 12 |- ((uxv /\ x e. Poset) -> (x infw {u, v}) e. dom x)
23	14, 22	jca 310	. . . . . . . . . . 11 |- ((uxv /\ x e. Poset) -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x))
24	23	ex 402	. . . . . . . . . 10 |- (uxv -> (x e. Poset -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
25		spwpr4c 10009	. . . . . . . . . . . . . . . . 17 |- ((x e. Poset /\ u e. _V /\ vxu) -> (x supw {v, u}) = u)
26	25	3exp 1066	. . . . . . . . . . . . . . . 16 |- (x e. Poset -> (u e. _V -> (vxu -> (x supw {v, u}) = u)))
27	19, 26	mpi 55	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (vxu -> (x supw {v, u}) = u))
28	27	impcom 378	. . . . . . . . . . . . . 14 |- ((vxu /\ x e. Poset) -> (x supw {v, u}) = u)
29	8	pre2befi2 14573	. . . . . . . . . . . . . . . . . 18 |- ((x e. Preset /\ u e. _V /\ vxu) -> u e. dom x)
30	29	3exp 1066	. . . . . . . . . . . . . . . . 17 |- (x e. Preset -> (u e. _V -> (vxu -> u e. dom x)))
31	19, 30	mpi 55	. . . . . . . . . . . . . . . 16 |- (x e. Preset -> (vxu -> u e. dom x))
32	7, 31	syl 12	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (vxu -> u e. dom x))
33	32	impcom 378	. . . . . . . . . . . . . 14 |- ((vxu /\ x e. Poset) -> u e. dom x)
34	28, 33	eqeltrd 1971	. . . . . . . . . . . . 13 |- ((vxu /\ x e. Poset) -> (x supw {v, u}) e. dom x)
35		prcom 3097	. . . . . . . . . . . . . 14 |- {u, v} = {v, u}
36		opreq2 4890	. . . . . . . . . . . . . . 15 |- ({u, v} = {v, u} -> (x supw {u, v}) = (x supw {v, u}))
37	36	eleq1d 1963	. . . . . . . . . . . . . 14 |- ({u, v} = {v, u} -> ((x supw {u, v}) e. dom x <-> (x supw {v, u}) e. dom x))
38	35, 37	ax-mp 7	. . . . . . . . . . . . 13 |- ((x supw {u, v}) e. dom x <-> (x supw {v, u}) e. dom x)
39	34, 38	sylibr 217	. . . . . . . . . . . 12 |- ((vxu /\ x e. Poset) -> (x supw {u, v}) e. dom x)
40		nfwpr4c 14630	. . . . . . . . . . . . . . . . 17 |- ((x e. Poset /\ u e. _V /\ vxu) -> (x infw {v, u}) = v)
41	40	3exp 1066	. . . . . . . . . . . . . . . 16 |- (x e. Poset -> (u e. _V -> (vxu -> (x infw {v, u}) = v)))
42	19, 41	mpi 55	. . . . . . . . . . . . . . 15 |- (x e. Poset -> (vxu -> (x infw {v, u}) = v))
43	42	impcom 378	. . . . . . . . . . . . . 14 |- ((vxu /\ x e. Poset) -> (x infw {v, u}) = v)
44	2	breldm 4161	. . . . . . . . . . . . . . 15 |- (vxu -> v e. dom x)
45	44	adantr 425	. . . . . . . . . . . . . 14 |- ((vxu /\ x e. Poset) -> v e. dom x)
46	43, 45	eqeltrd 1971	. . . . . . . . . . . . 13 |- ((vxu /\ x e. Poset) -> (x infw {v, u}) e. dom x)
47	35	opreq2i 4893	. . . . . . . . . . . . 13 |- (x infw {u, v}) = (x infw {v, u})
48	46, 47	syl5eqel 1975	. . . . . . . . . . . 12 |- ((vxu /\ x e. Poset) -> (x infw {u, v}) e. dom x)
49	39, 48	jca 310	. . . . . . . . . . 11 |- ((vxu /\ x e. Poset) -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x))
50	49	ex 402	. . . . . . . . . 10 |- (vxu -> (x e. Poset -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
51	24, 50	jaoi 368	. . . . . . . . 9 |- ((uxv \/ vxu) -> (x e. Poset -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
52	51	com12 14	. . . . . . . 8 |- (x e. Poset -> ((uxv \/ vxu) -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
53	52	adantr 425	. . . . . . 7 |- ((x e. Poset /\ v e. dom x) -> ((uxv \/ vxu) -> ((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
54	53	ralimdvaa 2171	. . . . . 6 |- (x e. Poset -> (A.v e. dom x(uxv \/ vxu) -> A.v e. dom x((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
55	54	adantr 425	. . . . 5 |- ((x e. Poset /\ u e. dom x) -> (A.v e. dom x(uxv \/ vxu) -> A.v e. dom x((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
56	55	ralimdvaa 2171	. . . 4 |- (x e. Poset -> (A.u e. dom xA.v e. dom x(uxv \/ vxu) -> A.u e. dom xA.v e. dom x((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
57	56	imdistani 491	. . 3 |- ((x e. Poset /\ A.u e. dom xA.v e. dom x(uxv \/ vxu)) -> (x e. Poset /\ A.u e. dom xA.v e. dom x((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
58	8	istoset2 14628	. . 3 |- (x e. Toset <-> (x e. Poset /\ A.u e. dom xA.v e. dom x(uxv \/ vxu)))
59	8	isla 10010	. . 3 |- (x e. Lat <-> (x e. Poset /\ A.u e. dom xA.v e. dom x((x supw {u, v}) e. dom x /\ (x infw {u, v}) e. dom x)))
60	57, 58, 59	3imtr4i 236	. 2 |- (x e. Toset -> x e. Lat)
61	1, 60	mpgbir 1334	1 |- Toset C_ Lat

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {cpr 3045   class class class wbr 3338  dom cdm 3986  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981   inf_w cinf 9982  Latcla 9983   Toset ccha 10207   Preset cpreset 14555

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-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-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-reu 2111  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-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-opr 4886  df-oprab 4887  df-ps 9984  df-spw 9985  df-nfw 9986  df-la 9987  df-toset 10208  df-dir 10350  df-prs 14563

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