Nuprl Lemma : geo-line-eq-geoline
∀[e:EuclideanPlane]. ∀[P:LINE]. ∀[l:Line]. uiff(fst(l) I P ∧ fst(snd(l)) I P;l = P ∈ LINE)
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geoline: LINE,
geo-line: Line,
euclidean-plane: EuclideanPlane,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
geo-incident: p I L,
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-colinear: Colinear(a;b;c),
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
geoline: LINE,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
cand: A c∧ B,
basic-geometry: BasicGeometry
Lemmas referenced :
geo-incident_wf,
not_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
equal_wf,
geoline_wf,
subtype_quotient,
geo-line-eq_wf,
geo-line_wf,
geo-line-eq-equiv,
quotient-member-eq,
equal-wf-base,
geo-line-eq_functionality,
geo-line-eq_weakening,
geo-line-eq_inversion,
geo-incident-line,
pi1_wf_top,
geo-point_wf,
geo-colinear_wf,
geo-colinear-line-eq2,
and_wf,
subtype_rel_product,
geo-sep_wf,
top_wf,
squash_wf,
true_wf,
geo-colinear-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
productEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
sqequalRule,
hypothesis,
because_Cache,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
instantiate,
independent_isectElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaFormation,
addLevel,
impliesFunctionality,
isect_memberEquality,
voidEquality,
functionEquality,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
levelHypothesis,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[P:LINE]. \mforall{}[l:Line]. uiff(fst(l) I P \mwedge{} fst(snd(l)) I P;l = P)
Date html generated:
2018_05_22-PM-01_03_54
Last ObjectModification:
2018_05_10-PM-05_50_00
Theory : euclidean!plane!geometry
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