Nuprl Lemma : Dtri-iff-lsep
∀e:EuclideanPlane. ∀a,b,c:Point. (Dtri(e;a;b;c) ⇐⇒ a # bc)
Proof
Definitions occuring in Statement :
dist-tri: Dtri(g;a;b;c),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
dist-tri: Dtri(g;a;b;c),
cand: A c∧ B,
squash: ↓T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
true: True
Lemmas referenced :
dist-tri_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
Prop22-inequality-implies-triangle,
dist-iff-lt,
Euclid-Prop20_cycle,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-length-flip,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
productElimination,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (Dtri(e;a;b;c) \mLeftarrow{}{}\mRightarrow{} a \# bc)
Date html generated:
2019_10_16-PM-02_55_00
Last ObjectModification:
2019_02_27-AM-10_47_32
Theory : euclidean!plane!geometry
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