Nuprl Lemma : geo-add-length_functionality_wrt_le
∀[e:BasicGeometry]. ∀[x,y,x',y':Length]. (x + y ≤ x' + y') supposing (x ≤ x' and y ≤ y')
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-add-length: p + q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
respects-equality: respects-equality(S;T),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
uiff: uiff(P;Q),
false: False,
not: ¬A,
stable: Stable{P},
basic-geometry-: BasicGeometry-,
so_apply: x[s],
so_lambda: λ2x.t[x],
geo-add-length: p + q,
cand: A c∧ B,
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
quotient: x,y:A//B[x; y],
geo-length-type: Length,
prop: ℙ,
squash: ↓T,
geo-le: p ≤ q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
respects-equality-trivial,
respects-equality-sets,
geo-length-equiv,
respects-equality-quotient1,
geo-between-trivial,
geo-zero-length-iff,
geo-add-length-is-zero,
geo-add-length-implies-eq-zero,
geo-add-length-comm,
geo-add-length-assoc,
geo-add-length-cancel-left,
geo-congruent-iff-length,
istype-universe,
equal_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length-between,
geo-sep-O-X,
geo-between-same-side2,
istype-void,
stable__geo-between,
geo-between-exchange4,
basic-geometry-_wf,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-congruent_wf,
geo-extend_wf,
geo-Op-sep,
geo-sep_wf,
subtype_rel_sets_simple,
geo-extend-property,
geo-eq_weakening,
geo-between_functionality,
geo-X_wf,
geo-O_wf,
geo-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
geo-point_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-eq_wf,
iff_weakening_equal,
subtype_rel_self,
subtype-geo-length-type,
true_wf,
squash_wf,
geo-add-length_wf,
basic-geometry_wf,
geo-length-type_wf,
geo-le_wf
Rules used in proof :
setEquality,
dependent_pairFormation_alt,
voidElimination,
independent_pairFormation,
functionIsType,
dependent_set_memberEquality_alt,
setIsType,
rename,
setElimination,
applyLambdaEquality,
sqequalBase,
productIsType,
dependent_functionElimination,
equalityIstype,
because_Cache,
independent_functionElimination,
independent_isectElimination,
universeEquality,
instantiate,
natural_numberEquality,
lambdaEquality_alt,
applyEquality,
lambdaFormation_alt,
equalitySymmetry,
equalityTransitivity,
productElimination,
promote_hyp,
pertypeElimination,
pointwiseFunctionalityForEquality,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
isectElimination,
extract_by_obid,
universeIsType,
baseClosed,
thin,
hypothesisEquality,
imageMemberEquality,
hypothesis,
imageElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[x,y,x',y':Length]. (x + y \mleq{} x' + y') supposing (x \mleq{} x' and y \mleq{} y')
Date html generated:
2019_10_29-AM-09_14_52
Last ObjectModification:
2019_10_18-PM-03_17_24
Theory : euclidean!plane!geometry
Home
Index