Nuprl Lemma : hp-angle-sum-sep
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k:Point. (abc + xyz ≅ ijk ⇒ {i ≠ j ∧ j ≠ k})
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
guard: {T},
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
geo-out: out(p ab),
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
euclidean-plane-axioms,
geo-sep-sym,
hp-angle-sum_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,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
universeIsType,
inhabitedIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k:Point. (abc + xyz \mcong{} ijk {}\mRightarrow{} \{i \mneq{} j \mwedge{} j \mneq{} k\})
Date html generated:
2019_10_16-PM-02_05_49
Last ObjectModification:
2019_06_05-AM-09_37_11
Theory : euclidean!plane!geometry
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