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: supposing a all: x:A. B[x] exists: x:A. B[x] not: ¬A and: P ∧ Q equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] uimplies: supposing a member: t ∈ T not: ¬A implies:  Q false: False uall: [x:A]. B[x] euclidean-plane: EuclideanPlane prop: exists: x:A. B[x] and: P ∧ Q so_lambda: λ2x.t[x] so_apply: x[s] cand: 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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