Nuprl Lemma : geo-extend-exists1
∀e:BasicGeometry. ∀q,a,b,c:Point. (q ≠ a ⇒ (∃x:Point. ((b ≠ c ⇒ q-a-x) ∧ ax ≅ bc)))
Proof
Definitions occuring in Statement :
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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-strict-between_wf,
geo-sep_wf,
geo-extend-property1
Rules used in proof :
independent_isectElimination,
instantiate,
functionEquality,
productEquality,
independent_pairFormation,
setElimination,
dependent_pairFormation,
productElimination,
sqequalRule,
applyEquality,
isectElimination,
hypothesis,
because_Cache,
dependent_set_memberEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
rename,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}q,a,b,c:Point. (q \mneq{} a {}\mRightarrow{} (\mexists{}x:Point. ((b \mneq{} c {}\mRightarrow{} q-a-x) \mwedge{} ax \00D0 bc)))
Date html generated:
2017_10_02-PM-04_50_46
Last ObjectModification:
2017_08_05-AM-09_43_57
Theory : euclidean!plane!geometry
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