Nuprl Lemma : geo-extend_wf
∀e:EuclideanPlane. ∀q:Point. ∀a:{a:Point| q ≠ a} . ∀b,c:Point. (extend qa by bc ∈ {x:Point| q_a_x ∧ ax ≅ bc} )
Proof
Definitions occuring in Statement :
geo-extend: extend qa by bc,
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
and: P ∧ Q,
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]},
geo-extend: extend qa by bc,
geo-SCO: SCO(a;b;c;d),
pi1: fst(t),
geo-SC: SC(a;b;c;d),
record-select: r.x,
geo-extend-construction-ext,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s]
Lemmas referenced :
geo-extend-construction-ext,
all_wf,
euclidean-plane_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
sq_exists_wf,
geo-between_wf,
geo-congruent_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
hypothesisEquality,
introduction,
isectElimination,
independent_isectElimination,
setEquality,
because_Cache,
setElimination,
rename,
productEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}q:Point. \mforall{}a:\{a:Point| q \mneq{} a\} . \mforall{}b,c:Point.
(extend qa by bc \mmember{} \{x:Point| q\_a\_x \mwedge{} ax \00D0 bc\} )
Date html generated:
2017_10_02-PM-04_50_30
Last ObjectModification:
2017_08_09-PM-06_40_41
Theory : euclidean!plane!geometry
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