Nuprl Lemma : geo-line-eq_functionality
∀e:EuclideanPlane. ∀l,m,l1,m1:Line. (l ≡ l1 ⇒ m ≡ m1 ⇒ (l ≡ m ⇐⇒ l1 ≡ m1))
Proof
Definitions occuring in Statement :
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-line-eq_inversion,
geo-line-eq_transitivity,
geo-line_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-line-eq_wf
Rules used in proof :
independent_functionElimination,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}l,m,l1,m1:Line. (l \mequiv{} l1 {}\mRightarrow{} m \mequiv{} m1 {}\mRightarrow{} (l \mequiv{} m \mLeftarrow{}{}\mRightarrow{} l1 \mequiv{} m1))
Date html generated:
2018_05_22-PM-01_02_22
Last ObjectModification:
2018_05_21-AM-01_33_08
Theory : euclidean!plane!geometry
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