Nuprl Lemma : geo-le-iff-between-points
∀g:EuclideanPlane. ∀p,q:{p:Point| O_X_p} . (p ≤ q ⇐⇒ X_p_q)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-X: X,
geo-O: O,
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]}
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],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
geo-le: p ≤ q,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
geo-eq: a ≡ b,
not: ¬A,
geo-length-type: Length,
quotient: x,y:A//B[x; y],
false: False,
cand: A c∧ B
Lemmas referenced :
geo-le_wf,
subtype-geo-length-type,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-X_wf,
geo-point_wf,
geo-O_wf,
sq_stable__geo-between,
member_wf,
geo-eq_wf,
geo-sep_wf,
geo-between_functionality,
geo-eq_weakening,
geo-length-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
inhabitedIsType,
setIsType,
because_Cache,
imageElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
pertypeElimination,
productEquality,
setEquality,
dependent_pairFormation_alt,
productIsType,
equalityIsType1
Latex:
\mforall{}g:EuclideanPlane. \mforall{}p,q:\{p:Point| O\_X\_p\} . (p \mleq{} q \mLeftarrow{}{}\mRightarrow{} X\_p\_q)
Date html generated:
2019_10_16-PM-01_34_02
Last ObjectModification:
2018_10_03-AM-11_16_57
Theory : euclidean!plane!geometry
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