Nuprl Lemma : geo-add-length-cancel-left-lt2
∀e:BasicGeometry. ∀y,z:Length. (z < z + y ⇐⇒ 0 < y)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-zero-length: 0,
geo-length-type: Length,
basic-geometry: BasicGeometry,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
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],
prop: ℙ,
rev_implies: P ⇐ Q,
geo-zero-length: 0,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
geo-lt: p < q,
exists: ∃x:A. B[x],
cand: A c∧ B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
rev_uimplies: rev_uimplies(P;Q),
geo_ge: geo_ge(e; p; q)
Lemmas referenced :
geo-lt_wf,
geo-add-length_wf,
geo-zero-length_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-cancel-left-lt,
squash_wf,
true_wf,
geo-add-length-zero,
subtype_rel_self,
iff_weakening_equal,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-le_functionality_wrt_implies,
geo-le_weakening,
geo-add-length_functionality_wrt_le,
geo-add-length-comm,
geo-le-add1,
geo-le-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
inhabitedIsType,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
productElimination,
dependent_pairFormation_alt,
promote_hyp,
productIsType,
setElimination,
rename,
dependent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}y,z:Length. (z < z + y \mLeftarrow{}{}\mRightarrow{} 0 < y)
Date html generated:
2019_10_16-PM-01_20_09
Last ObjectModification:
2018_12_11-AM-11_15_34
Theory : euclidean!plane!geometry
Home
Index