Nuprl Lemma : geo-Aax2
∀g:EuclideanPlane. ∀a,b:Point. ∀l,m:Line. (a ≠ b ⇒ (a I l ∧ a I m) ⇒ (b I l ∧ b I m) ⇒ l ≡ m)
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geo-line-eq: l ≡ m,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
geo-line-eq: l ≡ m,
not: ¬A,
geo-line-sep: geo-line-sep(g;l;m),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
oriented-plane: OrientedPlane,
cand: A c∧ B,
or: P ∨ Q,
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 :
geo-line-sep_wf,
geo-sep_wf,
geo-point_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-line_wf,
geo-incident-line,
geo-incident_wf,
geoline-subtype1,
all_wf,
geo-line-eq_wf,
not-lsep-iff-colinear,
oriented-colinear-append,
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,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
dependent_pairEquality,
productEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
addLevel,
allFunctionality,
impliesFunctionality,
independent_pairFormation,
andLevelFunctionality,
lambdaEquality,
cumulativity,
universeEquality,
functionEquality,
independent_functionElimination,
voidElimination,
dependent_pairFormation,
inrFormation,
inlFormation,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b:Point. \mforall{}l,m:Line. (a \mneq{} b {}\mRightarrow{} (a I l \mwedge{} a I m) {}\mRightarrow{} (b I l \mwedge{} b I m) {}\mRightarrow{} l \mequiv{} m)
Date html generated:
2018_05_22-PM-01_19_01
Last ObjectModification:
2018_05_13-PM-11_21_37
Theory : euclidean!plane!geometry
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