Nuprl Lemma : basic-geo-axioms-imply
∀g:EuclideanPlaneStructure
(BasicGeometryAxioms(g) ⇒ ((∀a:Point. a ≡ a) ∧ (∀a,b:Point. ab ≅ ba) ∧ (∀a,b,c:Point. (a ≡ b ⇒ ac ≅ bc))))
Proof
Definitions occuring in Statement :
euclidean-plane-structure: EuclideanPlaneStructure,
basic-geo-axioms: BasicGeometryAxioms(g),
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
basic-geo-axioms: BasicGeometryAxioms(g),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
geo-eq: a ≡ b,
not: ¬A,
geo-sep: a # b,
guard: {T},
geo-ge: ab ≥ cd,
geo-congruent: ab ≅ cd,
geo-length-sep: ab # cd),
or: P ∨ Q,
false: False
Latex:
\mforall{}g:EuclideanPlaneStructure
(BasicGeometryAxioms(g)
{}\mRightarrow{} ((\mforall{}a:Point. a \mequiv{} a) \mwedge{} (\mforall{}a,b:Point. ab \mcong{} ba) \mwedge{} (\mforall{}a,b,c:Point. (a \mequiv{} b {}\mRightarrow{} ac \mcong{} bc))))
Date html generated:
2020_05_20-AM-09_43_03
Last ObjectModification:
2020_01_27-PM-10_38_10
Theory : euclidean!plane!geometry
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