Nuprl Lemma : eu-ab-eq-x
∀e:EuclideanPlane. ∀a,b:Point. ((a = b ∈ Point) ⇒ (X = |ab| ∈ {p:Point| O_X_p} ))
Proof
Definitions occuring in Statement :
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eu-length_wf,
eu-mk-seg_wf,
and_wf,
eu-between-eq-trivial-right,
iff_weakening_equal,
eu-O_wf,
eu-length-null-segment,
eu-X_wf,
eu-between-eq_wf,
euclidean-plane_wf,
eu-point_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
setEquality,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. ((a = b) {}\mRightarrow{} (X = |ab|))
Date html generated:
2016_05_18-AM-06_44_13
Last ObjectModification:
2016_01_16-PM-10_28_49
Theory : euclidean!geometry
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