Nuprl Lemma : geo-le_antisymmetry
∀e:BasicGeometry. ∀[p,q:Length]. (p = q ∈ Length) supposing (q ≤ p and p ≤ q)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
geo-length-type: Length,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
guard: {T},
so_apply: x[s1;s2],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
basic-geometry-: BasicGeometry-
Lemmas referenced :
subtype-geo-length-type,
quotient-member-eq,
geo-point_wf,
geo-between_wf,
geo-O_wf,
geo-X_wf,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
geo-length-equiv,
equal-wf-base,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-le_wf,
geo-length-type_wf,
geo-le-iff-between-points,
sq_stable__geo-eq,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange4,
subtype_rel_self,
basic-geometry-_wf,
geo-Op-sep,
geo-between-same,
geo-eq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
setEquality,
applyEquality,
dependent_functionElimination,
setElimination,
rename,
lambdaEquality_alt,
universeIsType,
instantiate,
independent_isectElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
setIsType,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_set_memberEquality_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}[p,q:Length]. (p = q) supposing (q \mleq{} p and p \mleq{} q)
Date html generated:
2019_10_16-PM-01_35_18
Last ObjectModification:
2018_10_03-PM-00_21_09
Theory : euclidean!plane!geometry
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