Nuprl Lemma : midpoints-preserve-congruence
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point.
((((a'=b'=c' ∧ a=b=c) ∧ a' ≠ c' ∧ a ≠ c) ∧ ac ≅ a'c') ⇒ (ab ≅ a'b' ∧ bc ≅ b'c'))
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry
Lemmas referenced :
geo-midpoint_wf,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent_wf,
geo-point_wf,
geo-bet-sep-cong-tri-exists,
midpoint-sep,
geo-congruent-iff-length,
geo-congruent-symmetry,
geo-congruent_functionality,
geo-eq_weakening,
geo-midpoint_functionality,
at-most-one-midpoint,
geo-length-flip,
geo-congruent-left-comm,
euclidean-plane-axioms,
geo-inner-three-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point.
((((a'=b'=c' \mwedge{} a=b=c) \mwedge{} a' \mneq{} c' \mwedge{} a \mneq{} c) \mwedge{} ac \00D0 a'c') {}\mRightarrow{} (ab \00D0 a'b' \mwedge{} bc \00D0 b'c'))
Date html generated:
2017_10_02-PM-06_35_03
Last ObjectModification:
2017_08_16-AM-11_24_57
Theory : euclidean!plane!geometry
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