Nuprl Lemma : out-implies-straightangle
∀e:EuclideanPlane. ∀a,b,c,a',b',c':Point. (a' ≠ b' ⇒ c' ≠ b' ⇒ out(b ac) ⇒ (abc ≅a a'b'c' ⇐⇒ out(b' a'c')))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-cong-angle: abc ≅a xyz,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
geo-out: out(p ab),
geo-midpoint: a=m=b,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-cong-angle_wf,
geo-out_wf,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-congruent-preserves-out,
geo-congruent-iff-length,
geo-length-flip,
geo-out_transitivity,
geo-between-out,
geo-between-sep,
geo-out_inversion,
geo-sep-sym,
symmetric-point-construction,
geo-proper-extend-exists,
midpoint-sep,
geo-strict-between-sep2,
geo-out-iff-between1,
geo-strict-between-sep3,
geo-strict-between-implies-between,
geo-between-symmetry,
zero-angles-congruent2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
rename
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',b',c':Point.
(a' \mneq{} b' {}\mRightarrow{} c' \mneq{} b' {}\mRightarrow{} out(b ac) {}\mRightarrow{} (abc \mcong{}\msuba{} a'b'c' \mLeftarrow{}{}\mRightarrow{} out(b' a'c')))
Date html generated:
2019_10_16-PM-01_56_05
Last ObjectModification:
2019_09_27-PM-03_55_31
Theory : euclidean!plane!geometry
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