Nuprl Lemma : geo-SCS-out
∀e:EuclideanPlane. ∀a:Point. ∀x,b:{b:Point| a ≠ b} . out(a bSCS(b;a;a;x))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-SCS: SCS(a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
basic-geometry: BasicGeometry,
geo-out: out(p ab),
not: ¬A,
basic-geometry-: BasicGeometry-,
false: False,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-SCS_wf,
sq_stable__geo-sep,
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-congruent_wf,
geo-SCO_wf,
geo-colinear_wf,
sq_stable__geo-out,
geo-congruent-sep,
not_wf,
equal_wf,
geo-simple-colinear-cases,
false_wf,
stable__false,
geo-between-symmetry,
geo-between-same-side,
geo-sep-sym,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-same,
geo-sep_functionality,
geo-eq_weakening,
geo-sep-irrefl',
geo-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
functionEquality,
productElimination,
setEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}x,b:\{b:Point| a \mneq{} b\} . out(a bSCS(b;a;a;x))
Date html generated:
2018_05_22-PM-00_00_24
Last ObjectModification:
2018_03_30-PM-06_16_08
Theory : euclidean!plane!geometry
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