Theorem laspwcl 10011

Description: Closure of the supremum (join) of two lattice elements.

Hypothesis

Ref	Expression
isla.1	|- X = dom R

Assertion

Ref	Expression
laspwcl	|- ((R e. Lat /\ A e. X /\ B e. X) -> (R supw {A, B}) e. X)

Proof of Theorem laspwcl

Step	Hyp	Ref	Expression
1		isla.1	. . . . 5 |- X = dom R
2	1	isla 10010	. . . 4 |- (R e. Lat <-> (R e. Poset /\ A.x e. X A.y e. X ((R supw {x, y}) e. X /\ (R infw {x, y}) e. X)))
3		simpl 346	. . . . . . 7 |- (((R supw {x, y}) e. X /\ (R infw {x, y}) e. X) -> (R supw {x, y}) e. X)
4	3	ralimi 2168	. . . . . 6 |- (A.y e. X ((R supw {x, y}) e. X /\ (R infw {x, y}) e. X) -> A.y e. X (R supw {x, y}) e. X)
5	4	ralimi 2168	. . . . 5 |- (A.x e. X A.y e. X ((R supw {x, y}) e. X /\ (R infw {x, y}) e. X) -> A.x e. X A.y e. X (R supw {x, y}) e. X)
6	5	adantl 424	. . . 4 |- ((R e. Poset /\ A.x e. X A.y e. X ((R supw {x, y}) e. X /\ (R infw {x, y}) e. X)) -> A.x e. X A.y e. X (R supw {x, y}) e. X)
7	2, 6	sylbi 216	. . 3 |- (R e. Lat -> A.x e. X A.y e. X (R supw {x, y}) e. X)
8		preq1 3098	. . . . . 6 |- (x = A -> {x, y} = {A, y})
9	8	opreq2d 4898	. . . . 5 |- (x = A -> (R supw {x, y}) = (R supw {A, y}))
10	9	eleq1d 1963	. . . 4 |- (x = A -> ((R supw {x, y}) e. X <-> (R supw {A, y}) e. X))
11		preq2 3099	. . . . . 6 |- (y = B -> {A, y} = {A, B})
12	11	opreq2d 4898	. . . . 5 |- (y = B -> (R supw {A, y}) = (R supw {A, B}))
13	12	eleq1d 1963	. . . 4 |- (y = B -> ((R supw {A, y}) e. X <-> (R supw {A, B}) e. X))
14	10, 13	rcla42v 2384	. . 3 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (R supw {x, y}) e. X -> (R supw {A, B}) e. X))
15	7, 14	mpan9 521	. 2 |- ((R e. Lat /\ (A e. X /\ B e. X)) -> (R supw {A, B}) e. X)
16	15	3impb 1063	1 |- ((R e. Lat /\ A e. X /\ B e. X) -> (R supw {A, B}) e. X)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {cpr 3045  dom cdm 3986  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981   inf_w cinf 9982  Latcla 9983

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-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

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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-la 9987

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