Nuprl Lemma : geo-colinear-congruence1
∀e:BasicGeometry. ∀A,B,C,P,Q:Point. (A ≠ B ⇒ Colinear(A;B;C) ⇒ AP ≅ AQ ⇒ BP ≅ BQ ⇒ CP ≅ CQ)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
geo-five-seg-compressed: FSC(a;b;c;d a';b';c';d'),
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
geo-colinear_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-congruent-refl,
geo-fsc-ap
Rules used in proof :
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,P,Q:Point. (A \mneq{} B {}\mRightarrow{} Colinear(A;B;C) {}\mRightarrow{} AP \00D0 AQ {}\mRightarrow{} BP \00D0 BQ {}\mRightarrow{} CP \00D0 CQ)
Date html generated:
2017_10_02-PM-06_31_50
Last ObjectModification:
2017_08_05-PM-04_42_46
Theory : euclidean!plane!geometry
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