Nuprl Lemma : eu-sum-eq-x
∀e:EuclideanPlane. ∀a,b,c,d:Point. ((X = |ab| + |cd| ∈ {p:Point| O_X_p} ) ⇒ ((a = b ∈ Point) ∧ (c = d ∈ Point)))
Proof
Definitions occuring in Statement :
eu-add-length: 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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
eu-add-length-comm,
eu-eq-x-implies-eq,
eu-length-null-segment,
eu-le-null-segment,
eu-le-add1,
iff_weakening_equal,
true_wf,
squash_wf,
eu-le_wf,
euclidean-plane_wf,
eu-mk-seg_wf,
eu-length_wf,
eu-add-length_wf,
eu-between-eq-trivial-right,
eu-X_wf,
eu-O_wf,
eu-between-eq_wf,
eu-point_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. ((X = |ab| + |cd|) {}\mRightarrow{} ((a = b) \mwedge{} (c = d)))
Date html generated:
2016_05_18-AM-06_44_08
Last ObjectModification:
2016_01_16-PM-10_28_59
Theory : euclidean!geometry
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