Nuprl Lemma : euclid-P1
∀e:EuclideanPlane. ∀A,B:Point. ∃C:Point. (AC=AB ∧ BC=AB ∧ AC=BC) supposing ¬(A = B ∈ Point)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-congruent: ab=cd,
eu-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
eu-point_wf,
not_wf,
equal_wf,
euclidean-plane_wf,
circle-circle-continuity1,
eu-extend-exists,
eu-between-eq_wf,
eu-congruent_wf,
exists_wf,
eu-between-eq-trivial-left,
eu-congruent-refl,
eu-congruent-iff-length,
eu-length-flip,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
equalityEquality,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
productElimination,
dependent_set_memberEquality,
productEquality,
dependent_pairFormation,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B:Point. \mexists{}C:Point. (AC=AB \mwedge{} BC=AB \mwedge{} AC=BC) supposing \mneg{}(A = B)
Date html generated:
2016_05_18-AM-06_45_52
Last ObjectModification:
2015_12_28-AM-09_22_54
Theory : euclidean!geometry
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