Nuprl Lemma : not-geo-lt
∀g:EuclideanPlane. ∀p,q:Length. (¬p < q ⇐⇒ q ≤ p)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-le: p ≤ q,
geo-length-type: Length,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
or: P ∨ Q,
stable: Stable{P},
uimplies: b supposing a,
geo-length-type: Length,
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
not_wf,
geo-lt_wf,
geo-le_wf,
geo-length-type_wf,
euclidean-plane_wf,
stable__geo-le,
false_wf,
or_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
equal-wf-base,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane-structure_wf,
geo-primitives_wf,
not-geo-lt-points
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
inhabitedIsType,
dependent_functionElimination,
functionEquality,
unionIsType,
functionIsType,
because_Cache,
independent_isectElimination,
unionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
productEquality,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
setElimination,
rename
Latex:
\mforall{}g:EuclideanPlane. \mforall{}p,q:Length. (\mneg{}p < q \mLeftarrow{}{}\mRightarrow{} q \mleq{} p)
Date html generated:
2019_10_16-PM-01_37_40
Last ObjectModification:
2018_10_04-AM-11_18_24
Theory : euclidean!plane!geometry
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