Nuprl Lemma : geo-SCS_wf
∀[g:EuclideanPlane]
∀c,d,a:Point. ∀b:{b:Point| b ≠ a ∧ c_b_d} .
(SCS(a;b;c;d) ∈ {v:Point| cv ≅ cd ∧ (v_b_SCO(a;b;c;d) ∧ Colinear(a;b;v)) ∧ (b ≠ d ⇒ v ≠ SCO(a;b;c;d))} )
Proof
Definitions occuring in Statement :
geo-SCS: SCS(a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-SCO: SCO(a;b;c;d),
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
geo-SCS: SCS(a;b;c;d),
geo-SCO: SCO(a;b;c;d),
euclidean-plane: EuclideanPlane,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
geo-midpoint: a=m=b,
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
guard: {T},
cand: A c∧ B,
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
geo-SC_wf,
set_wf,
geo-point_wf,
geo-congruent_wf,
geo-between_wf,
geo-sep_wf,
sympoint_wf,
geo-sep-sym,
geo-between-sep,
geo-midpoint_wf,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange3,
equal_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent-symmetry,
geo-congruent-sep,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
exists_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-colinear_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
functionEquality,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
axiomEquality,
setEquality,
independent_pairFormation,
dependent_pairFormation,
inrFormation,
inlFormation,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[g:EuclideanPlane]
\mforall{}c,d,a:Point. \mforall{}b:\{b:Point| b \mneq{} a \mwedge{} c\_b\_d\} .
(SCS(a;b;c;d) \mmember{} \{v:Point|
cv \mcong{} cd \mwedge{} (v\_b\_SCO(a;b;c;d) \mwedge{} Colinear(a;b;v)) \mwedge{} (b \mneq{} d {}\mRightarrow{} v \mneq{} SCO(a;b;c;d))\} )
Date html generated:
2018_05_22-AM-11_54_58
Last ObjectModification:
2018_03_30-PM-06_00_51
Theory : euclidean!plane!geometry
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