Nuprl Lemma : geo-perp-in-iff
∀e:BasicGeometry. ∀a,b,c,d,x:Point.
(a ≠ b
⇒ c ≠ d
⇒ (ab ⊥x cd
⇐⇒ Colinear(a;b;x) ∧ Colinear(c;d;x) ∧ (∃u,v:Point. (Colinear(a;b;u) ∧ Colinear(c;d;v) ∧ u ≠ x ∧ v ≠ x ∧ Ruxv))))
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
right-angle: Rabc,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
geo-perp-in: ab ⊥x cd,
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
or: P ∨ Q,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs
Lemmas referenced :
right-angle_wf,
geo-sep_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
exists_wf,
geo-colinear_wf,
geo-perp-in_wf,
geo-sep-sym,
geo-colinear-same,
geo-sep-or,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-colinear-is-colinear-set,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
geo-colinear-append,
right-angle-colinear,
right-angle-symmetry
Rules used in proof :
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
because_Cache,
rename,
setElimination,
productEquality,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
unionElimination,
inrFormation,
inlFormation,
dependent_set_memberEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,x:Point.
(a \mneq{} b
{}\mRightarrow{} c \mneq{} d
{}\mRightarrow{} (ab \mbot{}x cd
\mLeftarrow{}{}\mRightarrow{} Colinear(a;b;x)
\mwedge{} Colinear(c;d;x)
\mwedge{} (\mexists{}u,v:Point. (Colinear(a;b;u) \mwedge{} Colinear(c;d;v) \mwedge{} u \mneq{} x \mwedge{} v \mneq{} x \mwedge{} Ruxv))))
Date html generated:
2017_10_02-PM-06_43_15
Last ObjectModification:
2017_08_05-PM-04_49_13
Theory : euclidean!plane!geometry
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