Nuprl Lemma : eu-conga-to-cong3
∀E:EuclideanPlane. ∀a,b,c,d,e,f:Point.
(abc = def ⇒ (∃a',c',d',f':Point. (out(b a'a) ∧ out(b cc') ∧ out(e d'd) ∧ out(e ff') ∧ Cong3(a'bc',d'ef'))))
Proof
Definitions occuring in Statement :
eu-out: out(p ab),
eu-cong-tri: Cong3(abc,a'b'c'),
eu-cong-angle: abc = xyz,
euclidean-plane: EuclideanPlane,
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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
eu-cong-angle: abc = xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
eu-out: out(p ab),
not: ¬A,
uimplies: b supposing a,
false: False,
eu-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q)
Lemmas referenced :
eu-cong-angle_wf,
eu-point_wf,
euclidean-plane_wf,
eu-out_wf,
eu-cong-tri_wf,
exists_wf,
eu-between-eq_wf,
eu-between-eq-same,
equal_wf,
not_wf,
eu-congruent_wf,
false_wf,
eu-congruence-identity,
eu-congruent-iff-length,
eu-length-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
dependent_pairFormation,
productEquality,
sqequalRule,
lambdaEquality,
because_Cache,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
independent_pairFormation,
dependent_functionElimination,
equalityTransitivity,
equalityEquality,
universeEquality
Latex:
\mforall{}E:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point.
(abc = def
{}\mRightarrow{} (\mexists{}a',c',d',f':Point. (out(b a'a) \mwedge{} out(b cc') \mwedge{} out(e d'd) \mwedge{} out(e ff') \mwedge{} Cong3(a'bc',d'ef'))))
Date html generated:
2016_10_26-AM-07_44_48
Last ObjectModification:
2016_07_12-AM-08_13_00
Theory : euclidean!geometry
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