Nuprl Lemma : geo-add-length-property2
∀g:EuclideanPlane. ∀s:geo-segment(g). ∀c,d:Point. cd ≅ |s||s| + |cd|
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
geo-segment: geo-segment(e),
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
geo-add-length: p + q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q)
Lemmas referenced :
geo-length_wf1,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-segment_wf,
geo-extend_wf,
geo-O_wf,
subtype_rel_sets,
geo-between_wf,
geo-X_wf,
geo-sep_wf,
geo-Op-sep,
geo-mk-seg_wf,
sq_stable__geo-congruent,
geo-length-property,
geo-congruent-iff-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
hypothesis,
inhabitedIsType,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
applyEquality,
instantiate,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaEquality_alt,
setIsType,
dependent_set_memberEquality_alt,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}s:geo-segment(g). \mforall{}c,d:Point. cd \mcong{} |s||s| + |cd|
Date html generated:
2019_10_16-PM-01_33_39
Last ObjectModification:
2018_10_02-PM-10_51_23
Theory : euclidean!plane!geometry
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