Nuprl Lemma : geo-add-length-lt-cancel-for-double
∀e:EuclideanPlane. ∀a,b:{a:Point| O_X_a} . (a + a < b + b ⇒ a < b)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-X: X,
geo-O: O,
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
euclidean-plane: EuclideanPlane,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
true: True,
false: False
Lemmas referenced :
geo-sep-iff-or-lt,
geo-add-length_wf1,
geo-lt_wf,
geo-add-length_wf,
subtype-geo-length-type,
geo-sep-or,
geo-between_wf,
geo-O_wf,
geo-X_wf,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-add-length-cancel-left-lt,
geo-lt-add1_2,
geo-add-length-lt-sep,
geo-length-equality,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-lt-sep,
geo-lt_transitivity,
geo-le_weakening-lt,
geo-lt-irrefl2,
geo-add-length-cancel-right-lt,
geo-add-length-comm,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
isectElimination,
hypothesis,
productElimination,
independent_functionElimination,
inlFormation_alt,
universeIsType,
applyEquality,
because_Cache,
setElimination,
rename,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
instantiate,
independent_isectElimination,
unionElimination,
inhabitedIsType,
setIsType,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
voidElimination,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:\{a:Point| O\_X\_a\} . (a + a < b + b {}\mRightarrow{} a < b)
Date html generated:
2019_10_16-PM-01_38_58
Last ObjectModification:
2019_03_20-PM-02_52_09
Theory : euclidean!plane!geometry
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