Nuprl Lemma : geo-eq-preserves-col
∀g:EuclideanPlane. ∀a,b,x,y:Point. (a ≡ b ⇒ Colinear(a;x;y) ⇒ Colinear(b;x;y))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-eq_weakening,
geo-colinear_functionality,
geo-point_wf,
geo-eq_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-colinear_wf,
geo-eq_inversion
Rules used in proof :
productElimination,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
sqequalRule,
instantiate,
applyEquality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,x,y:Point. (a \mequiv{} b {}\mRightarrow{} Colinear(a;x;y) {}\mRightarrow{} Colinear(b;x;y))
Date html generated:
2018_05_22-AM-11_54_05
Last ObjectModification:
2018_05_21-AM-01_13_30
Theory : euclidean!plane!geometry
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