Nuprl Lemma : geo-length-between-property
∀e:BasicGeometry. ∀p,q:{p:Point| O_X_p} . ∀a,b,c,d:Point.
(X_p_q ⇒ (p = |ab| ∈ Length) ⇒ (q = |cd| ∈ Length) ⇒ X_|ab|_|cd|)
Proof
Definitions occuring in Statement :
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
geo-le: p ≤ q,
exists: ∃x:A. B[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
squash: ↓T,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-le-iff-between-points,
subtype-geo-length-type,
geo-between_wf,
geo-X_wf,
geo-length_wf1,
geo-mk-seg_wf,
geo-length_wf,
geo-length-type_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-point_wf,
geo-O_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
dependent_pairFormation_alt,
because_Cache,
independent_pairFormation,
sqequalRule,
productIsType,
equalityIstype,
applyEquality,
isectElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
setElimination,
rename,
imageMemberEquality,
baseClosed,
inhabitedIsType,
instantiate,
independent_isectElimination,
setIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q:\{p:Point| O\_X\_p\} . \mforall{}a,b,c,d:Point.
(X\_p\_q {}\mRightarrow{} (p = |ab|) {}\mRightarrow{} (q = |cd|) {}\mRightarrow{} X\_|ab|\_|cd|)
Date html generated:
2019_10_16-PM-01_39_23
Last ObjectModification:
2019_02_28-PM-03_05_56
Theory : euclidean!plane!geometry
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