Theorem laspwcl 8930

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

Hypothesis

Ref	Expression
isla.1	⊢ X = dom R

Assertion

Ref	Expression
laspwcl	⊢ ((R ∈ Lat ⋀ A ∈ X ⋀ B ∈ X) → (R supw {A, B}) ∈ X)

Proof of Theorem laspwcl

Step	Hyp	Ref	Expression
1		isla.1	. . . . 5 ⊢ X = dom R
2	1	isla 8929	. . . 4 ⊢ (R ∈ Lat ↔ (R ∈ Poset ⋀ ∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))
3		pm3.26 317	. . . . . . 7 ⊢ (((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X) → (R supw {x, y}) ∈ X)
4	3	r19.20si 1752	. . . . . 6 ⊢ (∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X) → ∀y ∈ X (R supw {x, y}) ∈ X)
5	4	r19.20si 1752	. . . . 5 ⊢ (∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X) → ∀x ∈ X ∀y ∈ X (R supw {x, y}) ∈ X)
6	5	adantl 388	. . . 4 ⊢ ((R ∈ Poset ⋀ ∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)) → ∀x ∈ X ∀y ∈ X (R supw {x, y}) ∈ X)
7	2, 6	sylbi 197	. . 3 ⊢ (R ∈ Lat → ∀x ∈ X ∀y ∈ X (R supw {x, y}) ∈ X)
8		preq1 2509	. . . . . 6 ⊢ (x = A → {x, y} = {A, y})
9	8	opreq2d 4034	. . . . 5 ⊢ (x = A → (R supw {x, y}) = (R supw {A, y}))
10	9	eleq1d 1583	. . . 4 ⊢ (x = A → ((R supw {x, y}) ∈ X ↔ (R supw {A, y}) ∈ X))
11		preq2 2510	. . . . . 6 ⊢ (y = B → {A, y} = {A, B})
12	11	opreq2d 4034	. . . . 5 ⊢ (y = B → (R supw {A, y}) = (R supw {A, B}))
13	12	eleq1d 1583	. . . 4 ⊢ (y = B → ((R supw {A, y}) ∈ X ↔ (R supw {A, B}) ∈ X))
14	10, 13	rcla42v 1926	. . 3 ⊢ ((A ∈ X ⋀ B ∈ X) → (∀x ∈ X ∀y ∈ X (R supw {x, y}) ∈ X → (R supw {A, B}) ∈ X))
15	7, 14	mpan9 472	. 2 ⊢ ((R ∈ Lat ⋀ (A ∈ X ⋀ B ∈ X)) → (R supw {A, B}) ∈ X)
16	15	3impb 835	1 ⊢ ((R ∈ Lat ⋀ A ∈ X ⋀ B ∈ X) → (R supw {A, B}) ∈ X)

Syntax hints:   → wi 3   ⋀ wa 221   ⋀ w3a 781   = wceq 992   ∈ wcel 994  ∀wral 1691  {cpr 2468  dom cdm 3251  (class class class)co 4021  Posetcps 8895   sup_w cspw 8896   inf_w cinf 8897  Latcla 8900

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rab 1698  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-xp 3265  df-cnv 3267  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fv 3279  df-opr 4023  df-la 8906

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