Nuprl Lemma : geo-out-preserves-opp-side
∀e:BasicGeometry. ∀p,q,a,b,c,r,m:Point.
(p ≠ q ⇒ a-pq-c ⇒ Colinear(p;q;m) ⇒ a=m=c ⇒ Colinear(p;q;r) ⇒ out(r ab) ⇒ b-pq-c)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-opp-side: P-AB-Q,
geo-midpoint: a=m=b,
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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-out: out(p ab),
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
oriented-plane: OrientedPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-midpoint: a=m=b,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-opp-side: P-AB-Q
Lemmas referenced :
geo-out_wf,
geo-colinear_wf,
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-midpoint_wf,
geo-opp-side_wf,
geo-sep_wf,
geo-point_wf,
stable__geo-opp-side,
false_wf,
or_wf,
not_wf,
geo-between_wf,
minimal-double-negation-hyp-elim,
geo-out_functionality,
geo-eq_weakening,
geo-midpoint_functionality,
geo-opp-side_functionality,
minimal-not-not-excluded-middle,
geo-out-preserves-opp-side-case1,
geo-between_functionality,
geo-sep_functionality,
symmetric-point-construction,
geo-sep-sym,
geo-colinear_functionality,
geo-midpoint-diagonals-between,
geo-congruent-sep,
geo-midpoint-diagonals-congruent,
geo-midpoint-symmetry,
oriented-colinear-append,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
exists_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
geo-opp-side-sym,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry--subtype,
all_wf,
geo-between-symmetry,
geo-colinear-append,
geo-between-sep,
geo-between-outer-trans,
geo-out-colinear
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
independent_functionElimination,
unionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
promote_hyp,
impliesLevelFunctionality,
rename,
independent_pairFormation,
setEquality,
productEquality,
cumulativity,
dependent_pairFormation,
inrFormation,
inlFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,q,a,b,c,r,m:Point.
(p \mneq{} q {}\mRightarrow{} a-pq-c {}\mRightarrow{} Colinear(p;q;m) {}\mRightarrow{} a=m=c {}\mRightarrow{} Colinear(p;q;r) {}\mRightarrow{} out(r ab) {}\mRightarrow{} b-pq-c)
Date html generated:
2017_10_02-PM-06_45_35
Last ObjectModification:
2017_08_10-PM-06_06_02
Theory : euclidean!plane!geometry
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