Nuprl Lemma : geo-colinear-line-eq
∀e:EuclideanPlane. ∀a,b,c:Point. ∀l:{l:Line| a ≡ fst(l) ∧ b ≡ fst(snd(l))} . ∀m:{m:Line| b ≡ fst(m) ∧ c ≡ fst(snd(m))} .
(Colinear(a;b;c) ⇒ l ≡ m)
Proof
Definitions occuring in Statement :
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-point: Point,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
so_apply: x[s],
pi2: snd(t),
pi1: fst(t),
geo-line: Line,
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
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],
top: Top,
exists: ∃x:A. B[x],
false: False,
iff: P ⇐⇒ Q,
oriented-plane: OrientedPlane,
rev_implies: P ⇐ 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),
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
geo-eq_wf,
geo-line_wf,
set_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,
pi1_wf_top,
geo-eq_inversion,
geo-colinear_functionality,
geo-eq_weakening,
geo-lsep_functionality,
geo-sep_functionality,
lsep-not-between,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
lsep-all-sym,
colinear-lsep-cycle
Rules used in proof :
because_Cache,
productElimination,
productEquality,
lambdaEquality,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairEquality,
independent_functionElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. \mforall{}l:\{l:Line| a \mequiv{} fst(l) \mwedge{} b \mequiv{} fst(snd(l))\} . \mforall{}m:\{m:Line|
b \mequiv{} fst(m)
\mwedge{} c \mequiv{} fst(snd(m))\} .
(Colinear(a;b;c) {}\mRightarrow{} l \mequiv{} m)
Date html generated:
2018_05_22-PM-01_01_01
Last ObjectModification:
2018_01_16-PM-04_27_06
Theory : euclidean!plane!geometry
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