Nuprl Lemma : geo-lt-add1-iff
∀e:BasicGeometry. ∀p,q,r:{a:Point| O_X_a} . (X ≠ p ⇒ X ≠ q ⇒ X ≠ r ⇒ (p < q ⇐⇒ p + r < q + r))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
squash: ↓T,
true: True
Lemmas referenced :
geo-lt_wf,
subtype-geo-length-type,
geo-add-length_wf,
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-lt-add1_2,
geo-add-length-comm,
geo-add-length-cancel-left-lt,
squash_wf,
true_wf,
geo-length-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
instantiate,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
inhabitedIsType,
setIsType,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q,r:\{a:Point| O\_X\_a\} . (X \mneq{} p {}\mRightarrow{} X \mneq{} q {}\mRightarrow{} X \mneq{} r {}\mRightarrow{} (p < q \mLeftarrow{}{}\mRightarrow{} p + r < q + r))
Date html generated:
2019_10_16-PM-01_35_48
Last ObjectModification:
2019_01_16-AM-09_23_36
Theory : euclidean!plane!geometry
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