Nuprl Lemma : geo-zero-lt-iff
∀e:BasicGeometry. ∀u,v:Point. (0 < |uv| ⇐⇒ u ≠ v)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-zero-length: 0,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
geo-lt: p < q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
geo-zero-length: 0,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
rev_implies: P ⇐ Q,
cand: A c∧ B,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
geo-add-length-zero,
geo-sep_wf,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-sep-sym,
geo-length-flip,
iff_weakening_equal,
geo-le-same,
geo-add-length_wf,
geo-zero-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-comm,
subtype_rel_self,
geo-point_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-le-iff,
geo-ge-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
sqequalRule,
productIsType,
because_Cache,
universeIsType,
applyEquality,
hypothesis,
setElimination,
rename,
dependent_pairFormation_alt,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
inhabitedIsType,
promote_hyp,
instantiate,
universeEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}u,v:Point. (0 < |uv| \mLeftarrow{}{}\mRightarrow{} u \mneq{} v)
Date html generated:
2019_10_16-PM-01_19_46
Last ObjectModification:
2018_11_12-PM-03_20_39
Theory : euclidean!plane!geometry
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