Nuprl Lemma : eu-be-end-eq
∀e:EuclideanPlane. ∀a,b,c:Point. (a_b_c ⇒ ab=ac ⇒ (b = c ∈ Point))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
eu-congruence-identity-sym,
eu-length-null-segment,
eu-between-eq-symmetry,
eu-between-eq-trivial-right,
eu-add-length-cancel-left,
eu-add-length-zero,
iff_weakening_equal,
eu-mk-seg_wf,
eu-length_wf,
true_wf,
squash_wf,
eu-add-length_wf,
eu-X_wf,
eu-O_wf,
eu-congruent-iff-length,
eu-add-length-between,
euclidean-plane_wf,
eu-point_wf,
eu-between-eq_wf,
eu-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
productElimination,
equalityEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a\_b\_c {}\mRightarrow{} ab=ac {}\mRightarrow{} (b = c))
Date html generated:
2016_05_18-AM-06_45_35
Last ObjectModification:
2016_01_16-PM-10_29_28
Theory : euclidean!geometry
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