Nuprl Lemma : p4eu
∀e:EuclideanPlane. ∀a,b,c,A,B,C:Point.
(ab=AB ∧ ac=AC ∧ bac = BAC) ⇒ (bc=BC ∧ abc = ABC ∧ bca = BCA ∧ Cong3(abc,ABC))
supposing Triangle(a;b;c) ∧ Triangle(A;B;C)
Proof
Definitions occuring in Statement :
eu-cong-tri: Cong3(abc,a'b'c'),
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,
cand: A c∧ B,
prop: ℙ,
eu-cong-angle: abc = xyz,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
eu-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
eu-point_wf,
eu-congruent_wf,
eu-cong-angle_wf,
eu-tri_wf,
euclidean-plane_wf,
eu-sas,
equal_wf,
not_wf,
eu-congruence-identity,
false_wf,
eu-between-eq_wf,
exists_wf,
eu-between-eq-trivial-right,
eu-congruent-iff-length,
eu-length-flip
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,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
independent_pairFormation,
productEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
equalityTransitivity,
universeEquality,
dependent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,A,B,C:Point.
(ab=AB \mwedge{} ac=AC \mwedge{} bac = BAC) {}\mRightarrow{} (bc=BC \mwedge{} abc = ABC \mwedge{} bca = BCA \mwedge{} Cong3(abc,ABC))
supposing Triangle(a;b;c) \mwedge{} Triangle(A;B;C)
Date html generated:
2016_10_26-AM-07_46_19
Last ObjectModification:
2016_07_12-AM-08_16_54
Theory : euclidean!geometry
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