Nuprl Lemma : geo-congruent-preserves-strict-between
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point. (a-b-c ⇒ ab ≅ a'b' ⇒ ac ≅ a'c' ⇒ bc ≅ b'c' ⇒ a'-b'-c')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
geo-strict-between: a-b-c,
cand: A c∧ B,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-congruent-between-exists,
geo-strict-between-sep2,
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-strict-between-implies-between,
geo-congruent_wf,
geo-strict-between_wf,
geo-point_wf,
geo-congruent-refl,
geo-congruent-iff-length,
geo-length-flip,
geo-inner-five-segment,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep3,
geo-congruence-identity,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
productElimination,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point. (a-b-c {}\mRightarrow{} ab \mcong{} a'b' {}\mRightarrow{} ac \mcong{} a'c' {}\mRightarrow{} bc \mcong{} b'c' {}\mRightarrow{} a'-b'-c')
Date html generated:
2019_10_16-PM-01_17_33
Last ObjectModification:
2018_10_10-PM-03_25_22
Theory : euclidean!plane!geometry
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