Nuprl Lemma : geo-lt-add1-iff2
∀e:BasicGeometry. ∀a1,a2,b1,b2,c1,c2:Point. (|a1a2| < |b1b2| ⇐⇒ |a1a2| + |c1c2| < |b1b2| + |c1c2|)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
geo-eq: a ≡ b,
stable: Stable{P},
false: False,
not: ¬A,
or: P ∨ Q,
uiff: uiff(P;Q),
respects-equality: respects-equality(S;T),
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
geo-length-type: Length,
basic-geometry-: BasicGeometry-,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
geo-le: p ≤ q,
exists: ∃x:A. B[x],
geo-lt: p < q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-between-trivial,
not-lt-and-symm-le,
geo-add-length-cancel-right-lt,
geo-add-length_functionality_wrt_cong,
geo-congruent-symmetry,
geo-congruent-sep,
geo-lt-iff-strict-between-points,
geo-sep_functionality,
basic-geometry-_wf,
minimal-not-not-excluded-middle,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
istype-void,
geo-between-same-side-or,
not_wf,
false_wf,
stable__geo-between,
geo-add-length-between,
geo-congruent-iff-length,
geo-add-length-assoc,
respects-equality-set-trivial,
geo-length-equiv,
geo-eq_wf,
respects-equality-quotient1,
geo-sep-sym,
geo-between-outer-trans,
geo-between-exchange4,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-extend-exists,
geo-between-sep,
geo-le-add1,
geo-add-length-comm,
geo-le_wf,
iff_weakening_equal,
subtype_rel_self,
geo-length-equality,
istype-universe,
true_wf,
geo-le-sep,
sq_stable__geo-sep,
subtype-geo-length-type,
geo-length-type_wf,
equal_wf,
geo-X_wf,
geo-O_wf,
geo-between_wf,
squash_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
geo-add-length_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-lt_wf
Rules used in proof :
promote_hyp,
voidElimination,
unionIsType,
functionIsType,
unionElimination,
functionEquality,
unionEquality,
equalityIstype,
applyLambdaEquality,
hyp_replacement,
natural_numberEquality,
dependent_set_memberEquality_alt,
universeEquality,
equalitySymmetry,
equalityTransitivity,
lambdaEquality_alt,
independent_functionElimination,
dependent_functionElimination,
setEquality,
productEquality,
productIsType,
baseClosed,
imageMemberEquality,
imageElimination,
dependent_pairFormation_alt,
productElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
inhabitedIsType,
because_Cache,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
universeIsType,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a1,a2,b1,b2,c1,c2:Point. (|a1a2| < |b1b2| \mLeftarrow{}{}\mRightarrow{} |a1a2| + |c1c2| < |b1b2| + |c1c2|)
Date html generated:
2019_10_29-AM-09_15_57
Last ObjectModification:
2019_10_18-PM-03_28_38
Theory : euclidean!plane!geometry
Home
Index