Nuprl Lemma : extend-using-SC
∀e:EuclideanPlane. ∀q,a,b:Point. (q ≠ a ⇒ (∃x:Point. (q_a_x ∧ ax ≅ ab)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
geo-SCO_wf,
geo-sep-sym,
geo-between-trivial2,
geo-sep_wf,
geo-between_wf,
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-colinear_wf,
geo-congruent_wf,
sq_stable__geo-between,
equal_wf,
sq_stable__geo-congruent
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}q,a,b:Point. (q \mneq{} a {}\mRightarrow{} (\mexists{}x:Point. (q\_a\_x \mwedge{} ax \00D0 ab)))
Date html generated:
2017_10_02-PM-04_46_07
Last ObjectModification:
2017_08_08-PM-10_42_10
Theory : euclidean!plane!geometry
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