Nuprl Lemma : eu-colinear-cases
∀e:EuclideanStructure
∀[a,b,c:Point].
∀X:ℙ
(Stable{X}
⇒ (((¬(a = b ∈ Point)) ∧ (c = a ∈ Point)) ⇒ X)
⇒ (((¬(a = b ∈ Point)) ∧ (c = b ∈ Point)) ⇒ X)
⇒ (((¬(a = b ∈ Point)) ∧ c-a-b) ⇒ X)
⇒ (((¬(a = b ∈ Point)) ∧ a-c-b) ⇒ X)
⇒ (((¬(a = b ∈ Point)) ∧ a-b-c) ⇒ X)
⇒ ((¬Colinear(a;b;c)) ⇒ X)
⇒ X)
Proof
Definitions occuring in Statement :
eu-colinear: Colinear(a;b;c),
eu-between: a-b-c,
eu-point: Point,
euclidean-structure: EuclideanStructure,
stable: Stable{P},
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
stable: Stable{P},
uimplies: b supposing a,
not: ¬A,
member: t ∈ T,
prop: ℙ,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q
Lemmas referenced :
not_wf,
eu-colinear_wf,
and_wf,
equal_wf,
eu-point_wf,
eu-between_wf,
stable_wf,
euclidean-structure_wf,
eu-not-not-colinear,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
independent_isectElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
functionEquality,
universeEquality,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
unionElimination,
independent_pairFormation
Latex:
\mforall{}e:EuclideanStructure
\mforall{}[a,b,c:Point].
\mforall{}X:\mBbbP{}
(Stable\{X\}
{}\mRightarrow{} (((\mneg{}(a = b)) \mwedge{} (c = a)) {}\mRightarrow{} X)
{}\mRightarrow{} (((\mneg{}(a = b)) \mwedge{} (c = b)) {}\mRightarrow{} X)
{}\mRightarrow{} (((\mneg{}(a = b)) \mwedge{} c-a-b) {}\mRightarrow{} X)
{}\mRightarrow{} (((\mneg{}(a = b)) \mwedge{} a-c-b) {}\mRightarrow{} X)
{}\mRightarrow{} (((\mneg{}(a = b)) \mwedge{} a-b-c) {}\mRightarrow{} X)
{}\mRightarrow{} ((\mneg{}Colinear(a;b;c)) {}\mRightarrow{} X)
{}\mRightarrow{} X)
Date html generated:
2016_05_18-AM-06_35_44
Last ObjectModification:
2015_12_28-AM-09_28_57
Theory : euclidean!geometry
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