Nuprl Lemma : geo-le_imp
∀e:BasicGeometry. ∀p,q:{p:Point| O_X_p} . X_p_q supposing p ≤ q
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-between: a_b_c,
geo-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
geo-le: p ≤ q,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-length-type: Length,
quotient: x,y:A//B[x; y],
cand: A c∧ B,
iff: P ⇐⇒ Q
Lemmas referenced :
subtype-geo-length-type,
sq_stable__geo-between,
geo-X_wf,
geo-le_wf,
set_wf,
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-between_wf,
geo-O_wf,
member_wf,
geo-eq_wf,
geo-between_functionality,
geo-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
independent_functionElimination,
imageElimination,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
pertypeElimination,
productEquality,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q:\{p:Point| O\_X\_p\} . X\_p\_q supposing p \mleq{} q
Date html generated:
2017_10_02-PM-04_52_27
Last ObjectModification:
2017_08_17-PM-01_39_32
Theory : euclidean!plane!geometry
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