Nuprl Lemma : adjacent-right-angles
∀e:BasicGeometry. ∀a,b,c,a':Point. (b ≠ c ⇒ Rabc ⇒ Ra'bc ⇒ Colinear(a;b;a'))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
right-angle: Rabc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
right-angle: Rabc,
geo-midpoint: a=m=b,
and: P ∧ Q,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q)
Lemmas referenced :
symmetric-point-construction,
geo-midpoint-symmetry,
right-angle_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_wf,
geo-point_wf,
upper-dimension-axiom,
geo-congruent-iff-length,
geo-length-flip,
geo-between-sep,
geo-sep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
rename,
because_Cache,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a':Point. (b \mneq{} c {}\mRightarrow{} Rabc {}\mRightarrow{} Ra'bc {}\mRightarrow{} Colinear(a;b;a'))
Date html generated:
2018_05_22-PM-00_02_42
Last ObjectModification:
2018_04_02-AM-11_09_13
Theory : euclidean!plane!geometry
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