Nuprl Lemma : geo-tar-opp-side-invariant
∀e:BasicGeometry. ∀A,B,P,Q,C,D:Point.
(C ≠ D ⇒ Colinear(C;P;Q) ⇒ Colinear(D;P;Q) ⇒ geo-tar-opp-side(e;A;B;P;Q) ⇒ geo-tar-opp-side(e;A;B;C;D))
Proof
Definitions occuring in Statement :
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
colinear-lsep-general,
euclidean-plane-subtype-oriented,
basic-geometry-subtype,
subtype_rel_transitivity,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
geo-sep_wf,
geo-lsep_wf,
geo-colinear-symmetry,
geo-sep-sym,
lsep-all-sym,
oriented-colinear-append,
basic-geometry_wf,
euclidean-plane_wf,
oriented-plane_wf,
cons_wf,
geo-point_wf,
nil_wf,
lsep-implies-sep,
cons_member,
l_member_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
istype-void,
list_ind_nil_lemma,
length_of_cons_lemma,
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-between_wf,
geo-tar-opp-side_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
productIsType,
universeIsType,
functionIsType,
independent_pairFormation,
dependent_pairFormation_alt,
inrFormation_alt,
inlFormation_alt,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
lambdaEquality_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,P,Q,C,D:Point.
(C \mneq{} D
{}\mRightarrow{} Colinear(C;P;Q)
{}\mRightarrow{} Colinear(D;P;Q)
{}\mRightarrow{} geo-tar-opp-side(e;A;B;P;Q)
{}\mRightarrow{} geo-tar-opp-side(e;A;B;C;D))
Date html generated:
2019_10_16-PM-01_21_16
Last ObjectModification:
2018_12_11-PM-00_19_19
Theory : euclidean!plane!geometry
Home
Index