Nuprl Lemma : geo-zero-angle-congruence-out
∀g:EuclideanPlane. ∀a,b,c,x,y:Point. (abc ≅a xyx ⇒ out(b ac))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
geo-congruent-preserves-out,
geo-congruent-iff-length,
geo-between-implies-out2,
geo-between-out-implies-out2,
geo-sep-sym,
geo-between_wf,
geo-sep_wf,
geo-cong-angle_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
isectElimination,
independent_isectElimination,
hypothesis,
equalitySymmetry,
inlFormation_alt,
independent_pairFormation,
productIsType,
universeIsType,
applyEquality,
inhabitedIsType,
instantiate
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y:Point. (abc \mcong{}\msuba{} xyx {}\mRightarrow{} out(b ac))
Date html generated:
2019_10_16-PM-01_27_59
Last ObjectModification:
2018_11_08-AM-11_51_00
Theory : euclidean!plane!geometry
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