Nuprl Lemma : geo-out-preserves-opp-side-case1
∀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) ⇒ r_b_a ⇒ r ≠ b ⇒ b-pq-c)
Proof
Definitions occuring in Statement :
geo-opp-side: P-AB-Q,
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
and: P ∧ Q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
false: False,
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
geo-opp-side: P-AB-Q,
geo-colinear: Colinear(a;b;c),
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
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,
rev_implies: P ⇐ Q,
cand: A c∧ B,
geo-strict-between: a-b-c
Lemmas referenced :
minimal-not-not-excluded-middle,
geo-opp-side_functionality,
geo-midpoint_functionality,
geo-eq_weakening,
geo-between_functionality,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
stable__geo-opp-side,
geo-point_wf,
geo-opp-side_wf,
geo-midpoint_wf,
geo-colinear_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
basic-geometry--subtype,
geo-sep_wf,
geo-strict-between_wf,
exists_wf,
geo-strict-between-implies-between,
geo-between-symmetry,
Error :not-not-inner-pasch,
all_wf,
geo-between-same,
geo-colinear_functionality,
geo-sep_functionality,
geo-strict-between_functionality,
lelt_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-colinear-is-colinear-set,
equal_wf,
l_member_wf,
cons_member,
geo-sep-sym,
nil_wf,
cons_wf,
geo-colinear-append,
stable__not,
geo-strict-between-implies-colinear,
geo-strict-between-sep3,
geo-midpoint-implies-between,
geo-strict-between-sym,
geo-midpoint-id2
Rules used in proof :
productElimination,
dependent_functionElimination,
voidElimination,
unionElimination,
independent_functionElimination,
functionEquality,
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
lambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
promote_hyp,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidEquality,
isect_memberEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
impliesLevelFunctionality,
levelHypothesis,
impliesFunctionality,
addLevel
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{} r\_b\_a {}\mRightarrow{} r \mneq{} b {}\mRightarrow{} b-pq-c)
Date html generated:
2017_10_02-PM-06_45_21
Last ObjectModification:
2017_08_05-PM-04_50_29
Theory : euclidean!plane!geometry
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