Nuprl Lemma : geo-parallel_functionality
∀e:EuclideanPlane. ∀a,b,c,d,a',b',c',d':Point.
(a ≡ a' ⇒ b ≡ b' ⇒ c ≡ c' ⇒ d ≡ d' ⇒ geo-parallel(e;a;b;c;d) ⇒ geo-parallel(e;a';b';c';d'))
Proof
Definitions occuring in Statement :
geo-parallel: geo-parallel(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
oriented-plane: OrientedPlane,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
geo-parallel: geo-parallel(e;a;b;c;d),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-lsep_functionality,
geo-colinear_functionality,
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-eq_wf,
geo-parallel_wf,
geo-colinear_wf,
geo-eq_inversion,
geo-eq_weakening,
geo-sep_functionality
Rules used in proof :
instantiate,
sqequalRule,
applyEquality,
hypothesis,
independent_isectElimination,
isectElimination,
independent_functionElimination,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,a',b',c',d':Point.
(a \mequiv{} a' {}\mRightarrow{} b \mequiv{} b' {}\mRightarrow{} c \mequiv{} c' {}\mRightarrow{} d \mequiv{} d' {}\mRightarrow{} geo-parallel(e;a;b;c;d) {}\mRightarrow{} geo-parallel(e;a';b';c';d'))
Date html generated:
2018_05_22-PM-00_13_33
Last ObjectModification:
2018_05_21-AM-01_18_58
Theory : euclidean!plane!geometry
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