Nuprl Lemma : geo-pt-swap-preserves-parallel
∀e:EuclideanPlane. ∀a,b,c,d:Point. ∀A:a ≠ b. ∀C:c ≠ d. ∀A':b ≠ a. (<a, b, A> || <c, d, C> ⇒ <b, a, A'> || <c, d, C>)
Proof
Definitions occuring in Statement :
geo-Aparallel: l || m,
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
geo-line: Line,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
geoline: LINE,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
basic-geometry: BasicGeometry
Lemmas referenced :
geo-Aparallel_wf,
squash_wf,
true_wf,
geoline_wf,
trivial-equal,
geo-sep_wf,
geo-point_wf,
geoline-subtype1,
iff_weakening_equal,
euclidean-plane_wf,
quotient-member-eq,
geo-line_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-line-eq_wf,
geo-line-eq-equiv,
geo-colinear-line-eq2,
geo-colinear-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
sqequalRule,
dependent_pairEquality,
productEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
instantiate
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. \mforall{}A:a \mneq{} b. \mforall{}C:c \mneq{} d. \mforall{}A':b \mneq{} a.
(<a, b, A> || <c, d, C> {}\mRightarrow{} <b, a, A'> || <c, d, C>)
Date html generated:
2018_05_22-PM-01_18_48
Last ObjectModification:
2018_05_11-PM-06_53_57
Theory : euclidean!plane!geometry
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