Nuprl Lemma : geo-add-length-cancel-left-le
∀[e:BasicGeometry]. ∀[x,y,z:Length]. x ≤ y supposing z + x ≤ z + y
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-le: p ≤ q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
squash: ↓T,
geo-le: p ≤ q,
guard: {T},
implies: P ⇒ Q,
so_apply: x[s],
basic-geometry: BasicGeometry,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
prop: ℙ,
geo-length-type: Length,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
geo-add-length: p + q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
false: False,
not: ¬A,
stable: Stable{P},
true: True,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
geo-length-type_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-eq_wf,
equal-wf-base,
equal_wf,
geo-add-length_wf,
geo-X_wf,
geo-O_wf,
geo-between_wf,
geo-point_wf,
set_wf,
subtype-geo-length-type,
geo-le_wf,
geo-between-exchange4,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-congruent_wf,
geo-sep_wf,
geo-Op-sep,
geo-extend-property,
geo-eq_weakening,
geo-between_functionality,
geo-add-length_wf1,
member_wf,
exists_wf,
and_wf,
geo-sep-O-X,
geo-between-same-side2,
not_wf,
stable__geo-between,
geo-add-length-is-zero,
geo-add-length-implies-eq-zero,
iff_weakening_equal,
geo-add-length-assoc,
true_wf,
squash_wf,
geo-congruent-iff-length,
geo-add-length-between,
geo-mk-seg_wf,
geo-length_wf,
geo-length-flip,
geo-zero-length-iff,
geo-between-trivial
Rules used in proof :
isect_memberEquality,
baseClosed,
imageMemberEquality,
imageElimination,
independent_isectElimination,
instantiate,
productEquality,
independent_functionElimination,
lambdaFormation,
rename,
setElimination,
dependent_functionElimination,
lambdaEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
applyEquality,
productElimination,
pertypeElimination,
hypothesis,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality,
setEquality,
dependent_pairFormation,
independent_pairFormation,
voidElimination,
universeEquality,
natural_numberEquality,
applyLambdaEquality
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[x,y,z:Length]. x \mleq{} y supposing z + x \mleq{} z + y
Date html generated:
2017_10_02-PM-06_18_28
Last ObjectModification:
2017_08_05-PM-04_13_08
Theory : euclidean!plane!geometry
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