Nuprl Lemma : interior-angles-unique2

e:EuclideanPlane. ∀a,b,c,d,b',c',p:Point.
  (c leftof ab
   leftof ab
   leftof ab
   out(a bb')
   out(a cc')
   b'_p_c'
   p ≠ c'
   bad < bac
   bap ≅a bad
   out(a dp))


Proof




Definitions occuring in Statement :  geo-lt-angle: abc < xyz geo-out: out(p ab) geo-cong-angle: abc ≅a xyz euclidean-plane: EuclideanPlane geo-left: leftof bc geo-between: a_b_c geo-sep: a ≠ b geo-point: Point all: x:A. B[x] implies:  Q
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q geo-cong-angle: abc ≅a xyz and: P ∧ Q member: t ∈ T guard: {T} exists: x:A. B[x] basic-geometry: BasicGeometry euclidean-plane: EuclideanPlane uall: [x:A]. B[x] prop: subtype_rel: A ⊆B uimplies: supposing a basic-geometry-: BasicGeometry- iff: ⇐⇒ Q uiff: uiff(P;Q) cand: c∧ B geo-tri: Triangle(a;b;c) rev_implies:  Q
Lemmas referenced :  left-implies-sep geo-proper-extend-exists geo-O_wf geo-X_wf geo-sep-O-X geo-cong-angle_wf geo-lt-angle_wf geo-sep_wf euclidean-plane-structure-subtype euclidean-plane-subtype subtype_rel_transitivity euclidean-plane_wf euclidean-plane-structure_wf geo-primitives_wf geo-between_wf geo-out_wf geo-left_wf geo-point_wf geo-sep-sym geo-strict-between-sep3 geo-left-out-2 geo-out-iff-between1 geo-between-symmetry geo-strict-between-implies-between Euclid-Prop7 geo-congruent-iff-length geo-cong-angle-refl geo-out_weakening geo-eq_weakening euclidean-plane-axioms geo-cong-angle-symm2 geo-congruent-refl out-preserves-angle-cong_1 geo-cong-angle-transitivity geo-sas geo-out_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut sqequalHypSubstitution productElimination thin introduction extract_by_obid dependent_functionElimination hypothesisEquality independent_functionElimination hypothesis sqequalRule setElimination rename because_Cache universeIsType isectElimination applyEquality instantiate independent_isectElimination inhabitedIsType dependent_set_memberEquality_alt equalitySymmetry independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,d,b',c',p:Point.
    (c  leftof  ab
    {}\mRightarrow{}  d  leftof  ab
    {}\mRightarrow{}  p  leftof  ab
    {}\mRightarrow{}  out(a  bb')
    {}\mRightarrow{}  out(a  cc')
    {}\mRightarrow{}  b'\_p\_c'
    {}\mRightarrow{}  p  \mneq{}  c'
    {}\mRightarrow{}  bad  <  bac
    {}\mRightarrow{}  bap  \mcong{}\msuba{}  bad
    {}\mRightarrow{}  out(a  dp))



Date html generated: 2019_10_16-PM-02_17_15
Last ObjectModification: 2019_03_14-PM-09_56_04

Theory : euclidean!plane!geometry


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