Nuprl Lemma : midpoint-construction_wf
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . (Mid(a;b) ∈ {d:Point| a=d=b} )
Proof
Definitions occuring in Statement :
midpoint-construction: Mid(a;b),
euclidean-plane: EuclideanPlane,
geo-midpoint: a=m=b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
Euclid-midpoint-1,
geo-CC-2,
use-plane-sep,
sq_stable__and,
geo-SS: geo-SS(g;a;b;u;v),
record-select: r.x,
midpoint-construction: Mid(a;b),
geo-CC: CC(a;b;c;d),
geo-CCR: geo-CCR(g;a;b;c;d),
geo-CCL: CCL(a;b;c;d),
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
Euclid-midpoint-1,
subtype_rel_self,
all_wf,
sq_exists_wf,
geo-midpoint_wf,
geo-CC-2,
use-plane-sep,
sq_stable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
because_Cache,
functionEquality,
cumulativity,
setEquality,
setElimination,
rename
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . (Mid(a;b) \mmember{} \{d:Point| a=d=b\} )
Date html generated:
2018_05_22-PM-00_08_02
Last ObjectModification:
2018_03_30-AM-10_58_03
Theory : euclidean!plane!geometry
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