Theorem isla 8929

Description: The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds.

Hypothesis

Ref	Expression
isla.1	⊢ X = dom R

Assertion

Ref	Expression
isla	⊢ (R ∈ Lat ↔ (R ∈ Poset ⋀ ∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))

Distinct variable groups:   x,y,R   x,X,y

Proof of Theorem isla

Step	Hyp	Ref	Expression
1		dmeq 3402	. . . 4 ⊢ (r = R → dom r = dom R)
2		isla.1	. . . 4 ⊢ X = dom R
3	1, 2	syl6eqr 1568	. . 3 ⊢ (r = R → dom r = X)
4		opreq1 4026	. . . . . 6 ⊢ (r = R → (r supw {x, y}) = (R supw {x, y}))
5	4, 3	eleq12d 1585	. . . . 5 ⊢ (r = R → ((r supw {x, y}) ∈ dom r ↔ (R supw {x, y}) ∈ X))
6		opreq1 4026	. . . . . 6 ⊢ (r = R → (r infw {x, y}) = (R infw {x, y}))
7	6, 3	eleq12d 1585	. . . . 5 ⊢ (r = R → ((r infw {x, y}) ∈ dom r ↔ (R infw {x, y}) ∈ X))
8	5, 7	anbi12d 631	. . . 4 ⊢ (r = R → (((r supw {x, y}) ∈ dom r ⋀ (r infw {x, y}) ∈ dom r) ↔ ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))
9	3, 8	raleq12d 1840	. . 3 ⊢ (r = R → (∀y ∈ dom r((r supw {x, y}) ∈ dom r ⋀ (r infw {x, y}) ∈ dom r) ↔ ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))
10	3, 9	raleq12d 1840	. 2 ⊢ (r = R → (∀x ∈ dom r∀y ∈ dom r((r supw {x, y}) ∈ dom r ⋀ (r infw {x, y}) ∈ dom r) ↔ ∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))
11		df-la 8906	. 2 ⊢ Lat = {r ∈ Poset∣∀x ∈ dom r∀y ∈ dom r((r supw {x, y}) ∈ dom r ⋀ (r infw {x, y}) ∈ dom r)}
12	10, 11	elrab2 1953	1 ⊢ (R ∈ Lat ↔ (R ∈ Poset ⋀ ∀x ∈ X ∀y ∈ X ((R supw {x, y}) ∈ X ⋀ (R infw {x, y}) ∈ X)))

Syntax hints:   ↔ wb 144   ⋀ wa 221   = 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 is referenced by:  laspwcl 8930  lanfwcl 8931  toplat 10884

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-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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