Nuprl Lemma : geo-perp-trivial-when-colinear
∀e:EuclideanPlane. ∀a,b,c,d:Point. (b ≠ a ⇒ Colinear(a;b;c) ⇒ ab ⊥d dc ⇒ d ≡ c)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
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,
geo-eq: a ≡ b,
not: ¬A,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
geo-perp: ab ⊥ cd,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-sep_wf,
geo-perp-in_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-sep-sym,
geo-perp-in-symmetry,
perp-col,
geo-perp-irrefl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
sqequalRule,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (b \mneq{} a {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} ab \mbot{}d dc {}\mRightarrow{} d \mequiv{} c)
Date html generated:
2019_10_16-PM-01_29_37
Last ObjectModification:
2018_12_11-PM-06_11_39
Theory : euclidean!plane!geometry
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