Nuprl Lemma : opp-side_half-plane-angle-congruence-lemma
∀e:EuclideanPlane. ∀b,p,b',p',a,c,a',c',d:Point.
((a leftof bp ∧ c leftof pb)
⇒ (a' leftof b'p' ∧ c' leftof p'b')
⇒ ((abp ≅a a'b'p' ∧ pbc ≅a p'b'c') ∧ (a-d-c ∧ out(b dp)) ∧ b ≠ d)
⇒ (∃a'',c'',d'':Point
((ba ≅ b'a'' ∧ (bc ≅ b'c'' ∧ bd ≅ b'd'') ∧ ad ≅ a''d'' ∧ dc ≅ d''c'' ∧ out(b' d''p')) ∧ a''_d''_c'')))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-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,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-strict-between: a-b-c,
iff: P ⇐⇒ Q,
basic-geometry-: BasicGeometry-,
oriented-plane: OrientedPlane
Lemmas referenced :
geo-cong-angle_wf,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
geo-sep_wf,
geo-left_wf,
geo-point_wf,
left-implies-sep,
geo-proper-extend-exists,
geo-sep-sym,
geo-strict-between-sep3,
geo-congruent-iff-length,
geo-congruent_wf,
exists_wf,
geo-out-iff-between1,
geo-between-symmetry,
euclidean-plane-axioms,
geo-strict-between-implies-between,
geo-out_inversion,
geo-between_wf,
geo-out_weakening,
geo-eq_weakening,
geo-sas2,
out-preserves-angle-cong_1,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep2,
geo-five-segment,
geo-congruent-comm,
Euclid-Prop7,
left-between-implies-right1,
geo-left-out,
geo-left-out-2,
geo-left-out-1,
geo-left-out-3,
geo-congruent_functionality,
geo-sep_functionality,
geo-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
productEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
rename,
dependent_pairFormation,
independent_pairFormation,
equalitySymmetry,
lambdaEquality,
equalityTransitivity,
dependent_set_memberEquality,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}b,p,b',p',a,c,a',c',d:Point.
((a leftof bp \mwedge{} c leftof pb)
{}\mRightarrow{} (a' leftof b'p' \mwedge{} c' leftof p'b')
{}\mRightarrow{} ((abp \mcong{}\msuba{} a'b'p' \mwedge{} pbc \mcong{}\msuba{} p'b'c') \mwedge{} (a-d-c \mwedge{} out(b dp)) \mwedge{} b \mneq{} d)
{}\mRightarrow{} (\mexists{}a'',c'',d'':Point
((ba \mcong{} b'a'' \mwedge{} (bc \mcong{} b'c'' \mwedge{} bd \mcong{} b'd'') \mwedge{} ad \mcong{} a''d'' \mwedge{} dc \mcong{} d''c'' \mwedge{} out(b' d''p'))
\mwedge{} a''\_d''\_c'')))
Date html generated:
2018_05_22-PM-00_20_03
Last ObjectModification:
2018_04_21-PM-10_36_54
Theory : euclidean!plane!geometry
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