Nuprl Lemma : geo-gt-implies-lt
∀g:EuclideanPlane. ∀a,b,c,d:Point. (ab > cd ⇒ (¬¬|cd| < |ab|))
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,
geo-gt: cd > ab,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-lt: p < q,
member: t ∈ T,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uiff: uiff(P;Q),
euclidean-plane: EuclideanPlane,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-add-length-between,
geo-congruent-iff-length,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
equal_wf,
geo-length-type_wf,
geo-le_wf,
iff_weakening_equal,
geo-le-same,
geo-sep_wf,
geo-lt_wf,
istype-void,
geo-gt_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalHypSubstitution,
imageElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
hypothesisEquality,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality_alt,
equalityTransitivity,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
hyp_replacement,
applyEquality,
lambdaEquality_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeIsType,
voidElimination,
functionIsType,
instantiate
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab > cd {}\mRightarrow{} (\mneg{}\mneg{}|cd| < |ab|))
Date html generated:
2019_10_16-PM-01_19_16
Last ObjectModification:
2019_06_18-AM-10_56_10
Theory : euclidean!plane!geometry
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