Nuprl Lemma : geo-extend-property1
∀e:BasicGeometry. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c,d:Point. (bextend ab by cd ≅ cd ∧ (c ≠ d ⇒ a-b-extend ab by cd))
Proof
Definitions occuring in Statement :
geo-extend: extend qa by bc,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
implies: P ⇒ Q,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
geo-strict-between: a-b-c
Lemmas referenced :
sq_stable__geo-strict-between,
sq_stable__all,
sq_stable__geo-congruent,
sq_stable__and,
geo-strict-between_wf,
equal_wf,
geo-congruent_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
set_wf,
geo-sep_wf,
geo-congruent-sep,
sq_stable__geo-sep
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
functionEquality,
isect_memberEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
productEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
sqequalRule,
because_Cache,
applyEquality,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c,d:Point.
(bextend ab by cd \00D0 cd \mwedge{} (c \mneq{} d {}\mRightarrow{} a-b-extend ab by cd))
Date html generated:
2017_10_02-PM-04_50_38
Last ObjectModification:
2017_08_05-AM-09_43_52
Theory : euclidean!plane!geometry
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