Nuprl Lemma : geo-gt_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2,d1,d2:Point.
(a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ d1 ≡ d2 ⇒ (a1b1 > c1d1 ⇐⇒ a2b2 > c2d2))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-gt: cd > ab,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T
Lemmas referenced :
geo-eq_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
squash_wf,
exists_wf,
geo-between_wf,
geo-congruent_wf,
geo-sep_wf,
iff_wf,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-sep_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
instantiate,
independent_isectElimination,
independent_pairFormation,
lambdaEquality,
productEquality,
addLevel,
productElimination,
independent_functionElimination,
imageElimination,
existsFunctionality,
dependent_functionElimination,
andLevelFunctionality,
imageMemberEquality,
baseClosed,
impliesFunctionality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a1,a2,b1,b2,c1,c2,d1,d2:Point.
(a1 \mequiv{} a2 {}\mRightarrow{} b1 \mequiv{} b2 {}\mRightarrow{} c1 \mequiv{} c2 {}\mRightarrow{} d1 \mequiv{} d2 {}\mRightarrow{} (a1b1 > c1d1 \mLeftarrow{}{}\mRightarrow{} a2b2 > c2d2))
Date html generated:
2018_05_22-AM-11_53_57
Last ObjectModification:
2018_03_29-PM-06_34_58
Theory : euclidean!plane!geometry
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