Nuprl Lemma : eu-nontrivial_wf
∀[e:EuclideanStructure]
(eu-nontrivial(e) ∈ {triple:Point × Point × Point| let a,b,c = triple in (¬(a = b ∈ Point)) ∧ (¬Colinear(a;b;c))} )
Proof
Definitions occuring in Statement :
eu-nontrivial: eu-nontrivial(e),
eu-colinear: Colinear(a;b;c),
eu-point: Point,
euclidean-structure: EuclideanStructure,
spreadn: spread3,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eu-nontrivial: eu-nontrivial(e),
eu-colinear: Colinear(a;b;c),
eu-point: Point,
euclidean-structure: EuclideanStructure,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
prop: ℙ,
spreadn: spread3,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_rel_self,
not_wf,
equal_wf,
uall_wf,
iff_wf,
and_wf,
isect_wf,
euclidean-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
thin,
hypothesis,
applyEquality,
tokenEquality,
instantiate,
lemma_by_obid,
isectElimination,
universeEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
cumulativity,
hypothesisEquality,
because_Cache,
setEquality,
productEquality,
productElimination,
setElimination,
rename,
lambdaFormation,
axiomEquality
Latex:
\mforall{}[e:EuclideanStructure]
(eu-nontrivial(e) \mmember{} \{triple:Point \mtimes{} Point \mtimes{} Point|
let a,b,c = triple in
(\mneg{}(a = b)) \mwedge{} (\mneg{}Colinear(a;b;c))\} )
Date html generated:
2016_05_18-AM-06_32_31
Last ObjectModification:
2015_12_28-AM-09_28_40
Theory : euclidean!geometry
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