Nuprl Lemma : geo-le_witness
∀[e:BasicGeometry]. ∀[p,q:{p:Point| O_X_p} ]. Ax ∈ p ≤ q supposing X_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,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
geo-le: p ≤ q,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
and: P ∧ Q
Lemmas referenced :
subtype-geo-length-type,
geo-between_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-point_wf,
geo-O_wf,
equal_wf,
geo-length-type_wf,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
applyEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
setIsType,
because_Cache,
dependent_pairFormation_alt,
setEquality,
lambdaEquality,
productEquality,
independent_pairFormation,
dependent_pairFormation,
productIsType,
equalityIsType1
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[p,q:\{p:Point| O\_X\_p\} ]. Ax \mmember{} p \mleq{} q supposing X\_p\_q
Date html generated:
2019_10_16-PM-01_16_18
Last ObjectModification:
2018_11_08-PM-02_12_35
Theory : euclidean!plane!geometry
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