Nuprl Lemma : geo-colinear-line-eq2
∀e:EuclideanPlane. ∀l1,l2:Line.
(Colinear(fst(l1);fst(l2);fst(snd(l2))) ⇒ Colinear(fst(snd(l1));fst(l2);fst(snd(l2))) ⇒ l1 ≡ l2)
Proof
Definitions occuring in Statement :
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
or: P ∨ Q,
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
top: Top,
geo-line: Line,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
member: t ∈ T,
geo-line-sep: geo-line-sep(g;l;m),
not: ¬A,
geo-line-eq: l ≡ m,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
oriented-plane: OrientedPlane,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
false: False,
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),
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-colinear: Colinear(a;b;c)
Lemmas referenced :
geo-line_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-colinear_wf,
geo-line-sep_wf,
geo-sep_wf,
geo-point_wf,
pi1_wf_top,
geo-sep-or,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
lsep-implies-sep,
nil_wf,
cons_wf,
oriented-colinear-append,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-sep-sym,
lsep-all-sym,
colinear-lsep-cycle,
lsep-not-between,
geo-lsep_wf,
true_wf,
squash_wf,
top_wf,
subtype_rel_product,
and_wf,
colinear-lsep'
Rules used in proof :
independent_isectElimination,
instantiate,
unionElimination,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairEquality,
productElimination,
sqequalRule,
because_Cache,
applyEquality,
isectElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
productEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
independent_functionElimination,
levelHypothesis,
imageElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
addLevel
Latex:
\mforall{}e:EuclideanPlane. \mforall{}l1,l2:Line.
(Colinear(fst(l1);fst(l2);fst(snd(l2))) {}\mRightarrow{} Colinear(fst(snd(l1));fst(l2);fst(snd(l2))) {}\mRightarrow{} l1 \mequiv{} l2)
Date html generated:
2018_05_22-PM-01_01_14
Last ObjectModification:
2018_01_17-PM-00_01_56
Theory : euclidean!plane!geometry
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