Nuprl Lemma : lt-angle-not-cong
∀e:EuclideanPlane. ∀x,y,z:Point. (¬xyz < xyz)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A
Definitions unfolded in proof :
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
stable: Stable{P},
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-lt-angle: abc < xyz,
exists: ∃x:A. B[x],
geo-eq: a ≡ b,
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),
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
rev_implies: P ⇐ Q,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
geo-out: out(p ab)
Lemmas referenced :
geo-lt-angle_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
stable__not,
false_wf,
geo-lsep_wf,
not_wf,
istype-void,
minimal-double-negation-hyp-elim,
not-lsep-iff-colinear,
minimal-not-not-excluded-middle,
stable__false,
geo-sep_wf,
geo-sep_functionality,
geo-eq_weakening,
geo-between_functionality,
out-preserves-lsep,
lsep-all-sym,
colinear-lsep-cycle,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
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-between-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-congruent-refl,
geo-construction-unicity,
geo-sep-sym,
lsep-implies-sep,
geo-inner-three-segment,
geo-left_wf,
geo-congruent_functionality,
left-convex2,
geo-between_wf,
left-convex,
geo-out_transitivity,
geo-between-out,
geo-between-sep,
geo-out_inversion,
colinear-lsep,
geo-left-out-1,
geo-left-out-3,
Euclid-Prop7,
geo-left-out,
geo-left-out-2,
geo-out_wf,
geo-out_functionality,
geo-cong-angle_functionality,
geo-lsep_functionality,
out-preserves-angle-cong_1,
geo-zero-angle-congruence-out,
geo-colinear-cases,
geo-eq_wf,
geo-strict-between_wf,
euclidean-plane-axioms,
geo-between-trivial,
not-lt-zero-angle,
straight-angles-not-lt,
geo-strict-between-implies-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
inhabitedIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
unionEquality,
because_Cache,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
productElimination,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
equalityTransitivity,
equalitySymmetry,
inlFormation_alt,
inrFormation_alt,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}x,y,z:Point. (\mneg{}xyz < xyz)
Date html generated:
2019_10_16-PM-01_49_58
Last ObjectModification:
2019_09_27-PM-06_02_58
Theory : euclidean!plane!geometry
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