Nuprl Lemma : eu-le-null-segment
∀e:EuclideanPlane. ∀[p:{p:Point| O_X_p} ]. ∀[a:Point]. uiff(p ≤ |aa|;p = |aa| ∈ {p:Point| O_X_p} )
Proof
Definitions occuring in Statement :
eu-le: p ≤ q,
eu-length: |s|,
eu-mk-seg: ab,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
eu-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
euclidean-plane: EuclideanPlane,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
eu-le: p ≤ q,
sq_stable: SqStable(P)
Lemmas referenced :
eu-point_wf,
set_wf,
eu-between-eq_wf,
eu-O_wf,
eu-X_wf,
euclidean-plane_wf,
eu-le_wf,
eu-length_wf,
eu-mk-seg_wf,
uiff_wf,
eu-between-eq-trivial-right,
equal_wf,
squash_wf,
true_wf,
eu-length-null-segment,
iff_weakening_equal,
eu-between-eq-same,
sq_stable__eu-between-eq,
eu-between-eq-trivial-left
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
cumulativity,
dependent_set_memberEquality,
equalityEquality,
setEquality,
productElimination,
addLevel,
independent_pairFormation,
independent_isectElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
axiomEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[p:\{p:Point| O\_X\_p\} ]. \mforall{}[a:Point]. uiff(p \mleq{} |aa|;p = |aa|)
Date html generated:
2016_10_26-AM-07_42_03
Last ObjectModification:
2016_07_12-AM-08_08_20
Theory : euclidean!geometry
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