Nuprl Lemma : eu-eq-x-implies-eq
∀e:EuclideanPlane. ∀a,b:Point. ((X = |ab| ∈ {p:Point| O_X_p} ) ⇒ (a = b ∈ Point))
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,
eu-length: |s|,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
squash: ↓T,
true: True
Lemmas referenced :
eu-segment_wf,
and_wf,
euclidean-structure_wf,
true_wf,
squash_wf,
eu-congruent-iff-length,
eu_seg1_mk_seg_lemma,
eu_seg2_mk_seg_lemma,
eu-congruence-identity-sym,
eu-not-colinear-OXY,
eu-seg2_wf,
eu-seg1_wf,
eu-extend-equal-iff-congruent,
iff_weakening_equal,
eu-length-null-segment,
euclidean-plane_wf,
eu-mk-seg_wf,
eu-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,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
equalityEquality,
applyEquality,
lambdaEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. ((X = |ab|) {}\mRightarrow{} (a = b))
Date html generated:
2016_05_18-AM-06_44_00
Last ObjectModification:
2016_01_16-PM-10_28_53
Theory : euclidean!geometry
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