Nuprl Lemma : geo-be-end-eq
∀e:BasicGeometry. ∀a,b,c:Point. (a_b_c ⇒ ab ≅ ac ⇒ b ≡ c)
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
squash: ↓T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-zero-length: 0
Lemmas referenced :
geo-congruent_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-between_wf,
geo-point_wf,
geo-add-length-between,
geo-congruent-iff-length,
equal_wf,
squash_wf,
true_wf,
istype-universe,
geo-length-type_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-zero,
geo-add-length-cancel-left,
geo-zero-length_wf,
geo-length-null-segment,
geo-congruence-identity-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
productElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (a\_b\_c {}\mRightarrow{} ab \mcong{} ac {}\mRightarrow{} b \mequiv{} c)
Date html generated:
2019_10_16-PM-01_25_43
Last ObjectModification:
2018_12_15-PM-09_59_35
Theory : euclidean!plane!geometry
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