Nuprl Lemma : Euclid-Prop5
∀e:EuclideanPlane. ∀a,b,c,x,y:Point. (((ab ≅ ac ∧ a # bc) ∧ a-b-x ∧ a-c-y) ⇒ (abc ≅a acb ∧ xbc ≅a ycb))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-tri: Triangle(a;b;c),
geo-isosceles: ISOΔ(a;b;c)
Lemmas referenced :
geo-point_wf,
geo-strict-between_wf,
geo-lsep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-congruent_wf,
geo-sep-sym,
lsep-implies-sep,
Euclid-Prop5_1,
Euclid-Prop5_2
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
hypothesis,
independent_pairFormation,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y:Point.
(((ab \00D0 ac \mwedge{} a \# bc) \mwedge{} a-b-x \mwedge{} a-c-y) {}\mRightarrow{} (abc \00D0\msuba{} acb \mwedge{} xbc \00D0\msuba{} ycb))
Date html generated:
2017_10_02-PM-06_56_52
Last ObjectModification:
2017_08_10-PM-08_39_44
Theory : euclidean!plane!geometry
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