Nuprl Lemma : geo-between-congruent
∀e:BasicGeometry. ∀a,b,c:Point. (a_b_c ⇒ ac ≅ ab ⇒ c ≡ b)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
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],
and: P ∧ Q,
uiff: uiff(P;Q),
geo-zero-length: 0,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-add-length-between,
geo-congruent-iff-length,
geo-add-length-zero,
geo-zero-length_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length-cancel-left,
geo-eq_inversion,
geo-zero-length-iff
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
equalityTransitivity,
productElimination,
dependent_functionElimination,
rename,
setElimination,
independent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (a\_b\_c {}\mRightarrow{} ac \00D0 ab {}\mRightarrow{} c \mequiv{} b)
Date html generated:
2017_10_02-PM-06_15_06
Last ObjectModification:
2017_08_05-PM-04_11_55
Theory : euclidean!plane!geometry
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