Nuprl Lemma : geo-colinear-cases
∀e:BasicGeometry-
∀[a,b,c:Point].
∀X:ℙ
(Stable{X}
⇒ (a ≡ b ⇒ X)
⇒ (b ≡ c ⇒ X)
⇒ (c ≡ a ⇒ X)
⇒ (a-b-c ⇒ X)
⇒ (b-c-a ⇒ X)
⇒ (c-a-b ⇒ X)
⇒ ((¬Colinear(a;b;c)) ⇒ X)
⇒ X)
Proof
Definitions occuring in Statement :
basic-geometry-: BasicGeometry-,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-eq: a ≡ b,
geo-point: Point,
stable: Stable{P},
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
true: True,
or: P ∨ Q,
false: False,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
uimplies: b supposing a,
stable: Stable{P},
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
geo-eq: a ≡ b,
cand: A c∧ B,
and: P ∧ Q,
geo-strict-between: a-b-c
Lemmas referenced :
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
true_wf,
or_wf,
false_wf,
geo-point_wf,
stable_wf,
geo-eq_wf,
geo-strict-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry-_wf,
subtype_rel_transitivity,
basic-geometry--subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-colinear_wf,
not_wf,
geo-simple-colinear-cases,
geo-sep_wf,
stable__false,
geo-between_wf
Rules used in proof :
natural_numberEquality,
unionElimination,
voidElimination,
universeEquality,
because_Cache,
sqequalRule,
instantiate,
applyEquality,
functionEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation
Latex:
\mforall{}e:BasicGeometry-
\mforall{}[a,b,c:Point].
\mforall{}X:\mBbbP{}
(Stable\{X\}
{}\mRightarrow{} (a \mequiv{} b {}\mRightarrow{} X)
{}\mRightarrow{} (b \mequiv{} c {}\mRightarrow{} X)
{}\mRightarrow{} (c \mequiv{} a {}\mRightarrow{} X)
{}\mRightarrow{} (a-b-c {}\mRightarrow{} X)
{}\mRightarrow{} (b-c-a {}\mRightarrow{} X)
{}\mRightarrow{} (c-a-b {}\mRightarrow{} X)
{}\mRightarrow{} ((\mneg{}Colinear(a;b;c)) {}\mRightarrow{} X)
{}\mRightarrow{} X)
Date html generated:
2017_10_02-PM-04_43_44
Last ObjectModification:
2017_08_07-PM-00_25_05
Theory : euclidean!plane!geometry
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