### Nuprl Lemma : eu-le-null-segment

`∀e:EuclideanPlane. ∀[p:{p:Point| O_X_p} ]. ∀[a:Point].  uiff(p ≤ |aa|;p = |aa| ∈ {p:Point| O_X_p} )`

Proof

Definitions occuring in Statement :  eu-le: `p ≤ q` eu-length: `|s|` eu-mk-seg: `ab` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` euclidean-plane: `EuclideanPlane` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` implies: `P `` Q` rev_implies: `P `` Q` eu-le: `p ≤ q` sq_stable: `SqStable(P)`
Lemmas referenced :  eu-point_wf set_wf eu-between-eq_wf eu-O_wf eu-X_wf euclidean-plane_wf eu-le_wf eu-length_wf eu-mk-seg_wf uiff_wf eu-between-eq-trivial-right equal_wf squash_wf true_wf eu-length-null-segment iff_weakening_equal eu-between-eq-same sq_stable__eu-between-eq eu-between-eq-trivial-left
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis sqequalRule lambdaEquality because_Cache dependent_functionElimination natural_numberEquality cumulativity dependent_set_memberEquality equalityEquality setEquality productElimination addLevel independent_pairFormation independent_isectElimination applyEquality imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed universeEquality independent_functionElimination axiomEquality hyp_replacement Error :applyLambdaEquality

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}[p:\{p:Point|  O\_X\_p\}  ].  \mforall{}[a:Point].    uiff(p  \mleq{}  |aa|;p  =  |aa|)

Date html generated: 2016_10_26-AM-07_42_03
Last ObjectModification: 2016_07_12-AM-08_08_20

Theory : euclidean!geometry

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