Nuprl Lemma : geo-perp-in-symmetry2
∀e:BasicGeometry. ∀x:Point. ∀[a,b,c,d:Point]. (ab ⊥x cd ⇒ ba ⊥x cd)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
basic-geometry: BasicGeometry,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
geo-perp-in: ab ⊥x cd,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
sq_stable: SqStable(P),
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
prop: ℙ
Lemmas referenced :
sq_stable__colinear,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-colinear_wf,
geo-perp-in_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
imageElimination,
universeIsType,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}x:Point. \mforall{}[a,b,c,d:Point]. (ab \mbot{}x cd {}\mRightarrow{} ba \mbot{}x cd)
Date html generated:
2019_10_16-PM-01_29_05
Last ObjectModification:
2018_11_13-PM-00_25_06
Theory : euclidean!plane!geometry
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