Nuprl Lemma : geo-line-eq-to-col
∀g:EuclideanPlane. ∀l,m:Line.
(l ≡ m ⇒ {Colinear(fst(l);fst(snd(l));fst(m)) ∧ Colinear(fst(l);fst(snd(l));fst(snd(m)))})
Proof
Definitions occuring in Statement :
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
guard: {T},
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
pi2: snd(t),
pi1: fst(t),
geo-line: Line,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
and: P ∧ Q,
guard: {T},
geo-colinear: Colinear(a;b;c),
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry,
cand: A c∧ B,
exists: ∃x:A. B[x],
geo-line-sep: geo-line-sep(g;l;m),
geo-line-eq: l ≡ m,
oriented-plane: OrientedPlane,
rev_implies: P ⇐ Q,
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-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
Lemmas referenced :
euclidean-plane_wf,
geo-line_wf,
geo-line-eq_wf,
geo-lsep_wf,
or_wf,
false_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-between_wf,
not_wf,
stable__not,
minimal-double-negation-hyp-elim,
not-lsep-iff-colinear,
minimal-not-not-excluded-middle,
geo-colinear_wf,
geo-colinear-same,
oriented-colinear-append,
cons_wf,
geo-point_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
geo-sep_wf,
exists_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf
Rules used in proof :
dependent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalRule,
productElimination,
functionEquality,
because_Cache,
independent_isectElimination,
instantiate,
applyEquality,
productEquality,
independent_pairFormation,
independent_functionElimination,
unionElimination,
voidElimination,
dependent_pairFormation,
inrFormation,
inlFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:EuclideanPlane. \mforall{}l,m:Line.
(l \mequiv{} m {}\mRightarrow{} \{Colinear(fst(l);fst(snd(l));fst(m)) \mwedge{} Colinear(fst(l);fst(snd(l));fst(snd(m)))\})
Date html generated:
2018_05_22-PM-01_01_26
Last ObjectModification:
2018_01_17-PM-01_00_00
Theory : euclidean!plane!geometry
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