Nuprl Lemma : geo-lt-implies-gt
∀g:EuclideanPlane. ∀a,b,c,d:Point. (|cd| < |ab| ⇒ (¬¬ab > cd))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
or: P ∨ Q,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
geo-gt: cd > ab,
cand: A c∧ B,
geo-lt: p < q,
geo-zero-length: 0
Lemmas referenced :
not_wf,
geo-gt_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-point_wf,
stable__not,
false_wf,
or_wf,
geo-sep_wf,
minimal-double-negation-hyp-elim,
not_functionality_wrt_iff,
geo-gt_functionality,
geo-eq_weakening,
minimal-not-not-excluded-middle,
geo-proper-extend-exists,
geo-sep-sym,
geo-extend-exists,
geo-strict-between-sep3,
geo-between_functionality,
geo-congruent_functionality,
geo-out-le-iff-bet,
geo-out-iff-between1,
geo-congruent-sep,
geo-between-symmetry,
geo-congruent-iff-length,
squash_wf,
true_wf,
geo-length-type_wf,
subtype_rel_self,
iff_weakening_equal,
geo-le_weakening-lt,
geo-between_wf,
geo-congruent_wf,
geo-lt-not-congruent,
geo-lt-null-segment2,
geo-zero-length-iff,
basic-geometry_wf,
geo-between-trivial2,
geo-congruent-trivial,
geo-le-sep,
equal_wf,
geo-add-length-comm,
geo-add-length-zero,
geo-le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
because_Cache,
functionEquality,
unionElimination,
dependent_functionElimination,
productElimination,
independent_pairFormation,
promote_hyp,
equalitySymmetry,
lambdaEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairFormation,
productEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (|cd| < |ab| {}\mRightarrow{} (\mneg{}\mneg{}ab > cd))
Date html generated:
2019_10_16-PM-02_50_24
Last ObjectModification:
2018_09_16-PM-09_20_52
Theory : euclidean!plane!geometry
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