Nuprl Lemma : geo-zero-point-sep-iff-sep
∀e:BasicGeometry. ∀a,b:Point. (X ≠ |ab| ⇐⇒ a ≠ b)
Proof
Definitions occuring in Statement :
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-X: X,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
false: False
Lemmas referenced :
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-X_wf,
geo-length_wf1,
geo-mk-seg_wf,
geo-point_wf,
geo-sep-iff-or-lt,
geo-between-trivial,
geo-O_wf,
geo-between_wf,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-length-equality,
geo-length_wf,
subtype_rel_self,
iff_weakening_equal,
geo-lt-sep,
geo-lt-null-segment,
geo-zero-lt-iff,
geo-length-property,
trivial-zero-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
productElimination,
independent_functionElimination,
unionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
voidElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b:Point. (X \mneq{} |ab| \mLeftarrow{}{}\mRightarrow{} a \mneq{} b)
Date html generated:
2019_10_16-PM-01_38_07
Last ObjectModification:
2019_02_18-PM-07_47_10
Theory : euclidean!plane!geometry
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