Nuprl Lemma : geo-opp-side_functionality
∀[e:BasicGeometry]. ∀[A,B,P,Q,A',B',P',Q':Point]. (A ≡ A' ⇒ B ≡ B' ⇒ P ≡ P' ⇒ Q ≡ Q' ⇒ (P-AB-Q ⇐⇒ P'-A'B'-Q'))
Proof
Definitions occuring in Statement :
geo-opp-side: P-AB-Q,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
basic-geometry: BasicGeometry,
so_lambda: λ2x.t[x],
false: False,
not: ¬A,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
geo-opp-side: P-AB-Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
geo-colinear_wf,
not_wf,
geo-between_wf,
geo-point_wf,
all_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-eq_wf,
geo-opp-side_wf,
geo-colinear_functionality,
geo-eq_weakening,
geo-between_functionality
Rules used in proof :
voidElimination,
isect_memberEquality,
rename,
setElimination,
functionEquality,
independent_pairEquality,
productElimination,
dependent_functionElimination,
lambdaEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
andLevelFunctionality,
promote_hyp,
levelHypothesis,
impliesLevelFunctionality,
allLevelFunctionality,
independent_functionElimination,
allFunctionality,
impliesFunctionality,
addLevel
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[A,B,P,Q,A',B',P',Q':Point].
(A \mequiv{} A' {}\mRightarrow{} B \mequiv{} B' {}\mRightarrow{} P \mequiv{} P' {}\mRightarrow{} Q \mequiv{} Q' {}\mRightarrow{} (P-AB-Q \mLeftarrow{}{}\mRightarrow{} P'-A'B'-Q'))
Date html generated:
2017_10_02-PM-06_21_48
Last ObjectModification:
2017_08_05-PM-04_16_09
Theory : euclidean!plane!geometry
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