Nuprl Lemma : geo-lt-angle-construction
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(xyz < abc ⇒ x # yz ⇒ a # bc ⇒ (∃a',x',z':Point. (xya' ≅a abc ∧ x'-z'-a' ∧ out(y xx') ∧ out(y zz'))))
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-lsep: a # bc,
geo-strict-between: a-b-c,
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,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
squash: ↓T,
true: True,
geo-strict-between: a-b-c
Lemmas referenced :
cong-angle-out-exists1,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt-angle_wf,
geo-point_wf,
out-preserves-lsep,
lsep-all-sym,
colinear-lsep,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
geo-sep-sym,
lsep-implies-sep,
geo-cong-angle-symm2,
geo-between-sep,
geo-between_wf,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1,
geo-out_inversion,
geo-congruent-comm,
geo-out_wf,
geo-congruent_wf,
geo-congruent-iff-length,
geo-length-flip,
geo-extend-exists,
geo-five-segment,
geo-cong-angle_wf,
geo-strict-between_wf,
cong-tri-implies-cong-angle2,
geo-congruent-sep,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
universeIsType,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
functionIsType,
equalityTransitivity,
equalitySymmetry,
rename,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(xyz < abc
{}\mRightarrow{} x \# yz
{}\mRightarrow{} a \# bc
{}\mRightarrow{} (\mexists{}a',x',z':Point. (xya' \mcong{}\msuba{} abc \mwedge{} x'-z'-a' \mwedge{} out(y xx') \mwedge{} out(y zz'))))
Date html generated:
2019_10_16-PM-02_33_10
Last ObjectModification:
2019_09_24-PM-02_07_00
Theory : euclidean!plane!geometry
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