Nuprl Lemma : between-implies-straightangle
∀e:EuclideanPlane. ∀a,b,c,a',b',c':Point. (a-b-c ⇒ (abc ≅a a'b'c' ∧ a' ≠ b' ∧ b' ≠ c' ⇐⇒ a'-b'-c'))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: 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,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
rev_implies: P ⇐ Q,
geo-cong-angle: abc ≅a xyz,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
basic-geometry-: BasicGeometry-,
cand: A c∧ B,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
euclidean-plane: EuclideanPlane
Lemmas referenced :
geo-cong-angle_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-strict-between-sep2,
geo-strict-between-sep3,
geo-strict-between_wf,
geo-point_wf,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-between-exchange4,
geo-sep-sym,
geo-between-sep,
geo-congruent-preserves-between,
geo-congruent-iff-length,
geo-length-flip,
geo-between-inner-trans,
geo-between-exchange3,
cong-angle-between-exists-iff,
geo-proper-extend-exists,
geo-between_wf,
geo-congruent_wf,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
geo-length_wf,
geo-mk-seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
rename,
dependent_pairFormation_alt,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',b',c':Point.
(a-b-c {}\mRightarrow{} (abc \mcong{}\msuba{} a'b'c' \mwedge{} a' \mneq{} b' \mwedge{} b' \mneq{} c' \mLeftarrow{}{}\mRightarrow{} a'-b'-c'))
Date html generated:
2019_10_16-PM-01_56_19
Last ObjectModification:
2018_11_07-PM-01_05_02
Theory : euclidean!plane!geometry
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