Nuprl Lemma : geo-lt-add1
∀e:BasicGeometry. ∀p,q:{a:Point| O_X_a} . ∀r:{a:Point| O_X_a ∧ (|Xa| = p + q ∈ Length)} . (X ≠ p ⇒ X ≠ q ⇒ p < r)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
and: P ∧ Q,
basic-geometry-: BasicGeometry-,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
true: True,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
geo-lt: p < q,
cand: A c∧ B,
geo-le: p ≤ q,
geo-length-type: Length,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
respects-equality: respects-equality(S;T)
Lemmas referenced :
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-X_wf,
geo-point_wf,
geo-between_wf,
geo-O_wf,
geo-length-type_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
subtype-geo-length-type,
geo-construction-unicity-from-first,
subtype_rel_self,
basic-geometry-_wf,
geo-sep-O-X,
geo-strict-between-implies-between,
geo-between-symmetry,
sq_stable__geo-between,
geo-proper-extend-exists,
geo-between-outer-trans,
geo-between-exchange4,
geo-add-length-between,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
geo-congruent-iff-length,
geo-length-equality,
sq_stable__geo-congruent,
geo-strict-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-between-trivial,
respects-equality-quotient1,
geo-eq_wf,
geo-length-equiv,
respects-equality-sets,
respects-equality-trivial,
geo-le_wf,
geo-eq-self,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
setIsType,
productIsType,
equalityIstype,
inhabitedIsType,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
lambdaEquality_alt,
universeEquality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_pairFormation_alt,
setEquality,
productEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q:\{a:Point| O\_X\_a\} . \mforall{}r:\{a:Point| O\_X\_a \mwedge{} (|Xa| = p + q)\} .
(X \mneq{} p {}\mRightarrow{} X \mneq{} q {}\mRightarrow{} p < r)
Date html generated:
2019_10_16-PM-01_34_54
Last ObjectModification:
2019_01_10-PM-08_54_57
Theory : euclidean!plane!geometry
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