Nuprl Lemma : p5eu
∀e:EuclideanPlane. ∀a,b,c:Point. ((ab=ac ∧ Triangle(a;b;c)) ⇒ (∃j,k:Point. jbc = kcb))
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,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
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,
eu-tri: Triangle(a;b;c),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
eu-cong-angle: abc = xyz,
cand: A c∧ B,
not: ¬A,
false: False,
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
eu-congruent_wf,
eu-tri_wf,
eu-point_wf,
euclidean-plane_wf,
not_wf,
equal_wf,
eu-extend-exists,
eu-cong-angle_wf,
exists_wf,
eu-between-eq_wf,
eu-congruence-identity-sym,
false_wf,
eu-between-eq-trivial-right,
eu-congruent-iff-length,
eu-congruent-flip,
eu-congruent-refl,
eu-length-flip,
eu-three-segment,
eu-five-segment'
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
because_Cache,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
independent_functionElimination,
independent_isectElimination,
voidElimination,
equalityTransitivity,
equalityEquality,
universeEquality,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. ((ab=ac \mwedge{} Triangle(a;b;c)) {}\mRightarrow{} (\mexists{}j,k:Point. jbc = kcb))
Date html generated:
2016_10_26-AM-07_45_51
Last ObjectModification:
2016_07_12-AM-08_13_24
Theory : euclidean!geometry
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