Nuprl Lemma : Euclid-erect-perp-on-same-side
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:{c:Point| Colinear(a;b;c)} . ∀q:{q:Point| q # ab} .
(∃p:Point [(ab ⊥c pc ∧ p # ab ∧ (p leftof ab ⇐⇒ q leftof ab))])
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
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],
geo-lsep: a # bc,
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
basic-geometry: BasicGeometry,
not: ¬A,
false: False
Lemmas referenced :
sq_stable__geo-lsep,
Euclid-erect-2perp,
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-lsep_wf,
geo-colinear_wf,
geo-sep_wf,
lsep-all-sym2,
geo-left_wf,
geo-perp-in_wf,
iff_wf,
not-left-and-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
because_Cache,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
unionElimination,
dependent_set_memberFormation,
independent_pairFormation,
productEquality,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:\{c:Point| Colinear(a;b;c)\} . \mforall{}q:\{q:Point|
q \# ab\} .
(\mexists{}p:Point [(ab \mbot{}c pc \mwedge{} p \# ab \mwedge{} (p leftof ab \mLeftarrow{}{}\mRightarrow{} q leftof ab))])
Date html generated:
2018_05_22-PM-00_09_27
Last ObjectModification:
2018_04_04-PM-07_08_08
Theory : euclidean!plane!geometry
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