Nuprl Lemma : eu-sas
∀e:EuclideanPlane. ∀a,b,c,A,B,C:Point.
((ab=AB ∧ ac=AC) ∧ bac = BAC) ⇒ bc=BC supposing Triangle(a;b;c) ∧ Triangle(A;B;C)
Proof
Definitions occuring in Statement :
eu-cong-angle: abc = xyz,
eu-tri: Triangle(a;b;c),
euclidean-plane: EuclideanPlane,
eu-congruent: ab=cd,
eu-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
eu-tri: Triangle(a;b;c),
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
eu-cong-angle: abc = xyz,
exists: ∃x:A. B[x],
uiff: uiff(P;Q)
Lemmas referenced :
eu-inner-five-segment',
eu-five-segment',
eu-inner-three-segment,
eu-congruent-trivial,
equal_wf,
eu-length-flip,
eu-congruent-iff-length,
eu-between-eq-symmetry,
euclidean-plane_wf,
eu-tri_wf,
eu-cong-angle_wf,
eu-congruent_wf,
eu-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
equalityEquality,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
productEquality,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,A,B,C:Point.
((ab=AB \mwedge{} ac=AC) \mwedge{} bac = BAC) {}\mRightarrow{} bc=BC supposing Triangle(a;b;c) \mwedge{} Triangle(A;B;C)
Date html generated:
2016_06_16-PM-01_32_40
Last ObjectModification:
2016_06_01-PM-02_41_14
Theory : euclidean!geometry
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