Nuprl Lemma : erected-perp-opp-side-triangle
∀e:HeytingGeometry. ∀a,b,c,p,t:Point.
((c # ba ∧ ((ab ⊥ pa ∧ Colinear(a;b;t)) ∧ p-t-c) ∧ p # ba) ⇒ geo-tar-opp-side(e;p;c;a;b))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-perp: ab ⊥ cd,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
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,
and: P ∧ Q,
member: t ∈ T,
guard: {T},
cand: A c∧ B,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
geo-lsep: a # bc,
or: P ∨ Q,
geo-triangle: a # bc,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
basic-geometry: BasicGeometry
Lemmas referenced :
geo-triangle-symmetry,
geo-strict-between-implies-between,
subtype_rel_self,
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-colinear_wf,
geo-between_wf,
geo-triangle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
geo-primitives_wf,
geo-perp_wf,
geo-strict-between_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
independent_pairFormation,
dependent_pairFormation,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
productEquality,
cumulativity,
independent_isectElimination
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,p,t:Point.
((c \# ba \mwedge{} ((ab \mbot{} pa \mwedge{} Colinear(a;b;t)) \mwedge{} p-t-c) \mwedge{} p \# ba) {}\mRightarrow{} geo-tar-opp-side(e;p;c;a;b))
Date html generated:
2017_10_02-PM-07_08_03
Last ObjectModification:
2017_08_10-PM-04_53_56
Theory : euclidean!plane!geometry
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