Nuprl Lemma : lsep-implies-sep-or-lsep
∀e:EuclideanPlane. ∀a,b,c,x:Point. (a # bc ⇒ (c ≠ x ∨ x # ab))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
oriented-plane: OrientedPlane,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
subtype_rel: A ⊆r B,
guard: {T},
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
geo-equilateral: EQΔ(a;b;c),
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
cand: A c∧ B,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
iff: P ⇐⇒ Q
Lemmas referenced :
lsep-colinear-sep1,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
geo-colinear_wf,
lsep-symmetry,
euclidean-plane-axioms,
colinear-equidistant-points-exist,
geo-sep-sym,
lsep-implies-sep,
geo-sep_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
Euclid-midpoint,
sq_stable__midpoint,
midpoint-sep,
Euclid-Prop1,
upper-dimension-axiom,
geo-congruent-iff-length,
geo-length-flip,
colinear-lsep-cycle,
lsep-all-sym,
geo-between-implies-colinear,
geo-colinear-permute,
geo-colinear-append,
cons_wf,
nil_wf,
length_wf,
select_wf,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
l_member_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-sep-or,
colinear-lsep,
colinear-lsep',
lsep-symmetry2,
geo-colinear-cycle,
or_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
isectElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
universeIsType,
productIsType,
applyEquality,
because_Cache,
productElimination,
instantiate,
inhabitedIsType,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
int_eqEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x:Point. (a \# bc {}\mRightarrow{} (c \mneq{} x \mvee{} x \# ab))
Date html generated:
2019_10_16-PM-02_33_30
Last ObjectModification:
2019_07_23-PM-10_01_49
Theory : euclidean!plane!geometry
Home
Index