Nuprl Lemma : geo-perp-in-iff2
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ b ⇒ c ≠ d ⇒ (ab ⊥c cd ⇐⇒ Colinear(a;b;c) ∧ Racd ∧ Rbcd))
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
basic-geometry: BasicGeometry,
right-angle: Rabc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
geo-perp-in: ab ⊥x cd,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
or: P ∨ Q
Lemmas referenced :
geo-perp-in-iff,
geo-colinear-same,
geo-colinear_wf,
exists_wf,
geo-point_wf,
geo-sep_wf,
right-angle_wf,
geo-perp-in_wf,
iff_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep-or,
geo-sep-sym
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
sqequalRule,
isectElimination,
because_Cache,
productEquality,
applyEquality,
lambdaEquality,
addLevel,
impliesFunctionality,
instantiate,
independent_isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} b {}\mRightarrow{} c \mneq{} d {}\mRightarrow{} (ab \mbot{}c cd \mLeftarrow{}{}\mRightarrow{} Colinear(a;b;c) \mwedge{} Racd \mwedge{} Rbcd))
Date html generated:
2018_05_22-PM-00_05_14
Last ObjectModification:
2018_04_19-AM-01_40_34
Theory : euclidean!plane!geometry
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