Nuprl Lemma : geo-line-eq-preserves-incidence
∀g:EuclideanPlane. ∀a:Point. ∀l,m:Line. (a I l ⇒ l ≡ m ⇒ a I m)
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geo-line-eq: l ≡ m,
geo-line: Line,
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,
geoline: LINE,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
quotient-member-eq,
geo-line-eq_wf,
geo-line_wf,
geo-line-eq-equiv,
geo-incident_wf,
squash_wf,
true_wf,
geoline_wf,
subtype_rel_self,
iff_weakening_equal,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geoline-subtype1,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
hypothesisEquality,
applyEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a:Point. \mforall{}l,m:Line. (a I l {}\mRightarrow{} l \mequiv{} m {}\mRightarrow{} a I m)
Date html generated:
2018_05_22-PM-01_04_28
Last ObjectModification:
2018_05_11-PM-06_49_43
Theory : euclidean!plane!geometry
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