Nuprl Lemma : lsep-cong-angle-implies-sep
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (a # bc ⇒ abc ≅a xyz ⇒ x ≠ z)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
cong-angle-out-exists-cong3,
geo-congruent-sep,
geo-congruent-iff-length,
geo-length-flip,
out-preserves-lsep,
lsep-symmetry,
lsep-all-sym,
geo-out_inversion,
euclidean-plane-axioms,
geo-sep-sym,
lsep-implies-sep,
geo-cong-angle_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
because_Cache,
sqequalRule,
isectElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (a \# bc {}\mRightarrow{} abc \mcong{}\msuba{} xyz {}\mRightarrow{} x \mneq{} z)
Date html generated:
2019_10_16-PM-01_50_37
Last ObjectModification:
2018_11_19-PM-03_50_03
Theory : euclidean!plane!geometry
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