Nuprl Lemma : interior-angles-unique2-symm
∀e:EuclideanPlane. ∀a,b,c,d,a',c',p:Point.
(c leftof ab
⇒ d leftof ab
⇒ p leftof ab
⇒ out(b aa')
⇒ out(b cc')
⇒ a'_p_c'
⇒ p ≠ c'
⇒ abd < abc
⇒ abp ≅a abd
⇒ out(b 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: a leftof bc,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ 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: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
cand: A c∧ B,
geo-tri: Triangle(a;b;c),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ 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-3,
geo-out-if-between,
geo-strict-between-sym,
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-iff-between1,
geo-between-symmetry,
geo-strict-between-implies-between,
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,a',c',p:Point.
(c leftof ab
{}\mRightarrow{} d leftof ab
{}\mRightarrow{} p leftof ab
{}\mRightarrow{} out(b aa')
{}\mRightarrow{} out(b cc')
{}\mRightarrow{} a'\_p\_c'
{}\mRightarrow{} p \mneq{} c'
{}\mRightarrow{} abd < abc
{}\mRightarrow{} abp \mcong{}\msuba{} abd
{}\mRightarrow{} out(b dp))
Date html generated:
2019_10_16-PM-02_17_35
Last ObjectModification:
2019_08_12-PM-09_00_01
Theory : euclidean!plane!geometry
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