Nuprl Lemma : use-SC
∀e:EuclideanPlane. ∀a,b,c,d:Point.
(a ≠ b ⇒ c_b_d ⇒ (∃u,v:Point. (cu ≅ cd ∧ cv ≅ cd ∧ a_b_u ∧ (v_b_u ∧ Colinear(a;b;v)) ∧ (b ≠ d ⇒ v ≠ u))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
geo-colinear: Colinear(a;b;c),
not: ¬A,
false: False
Lemmas referenced :
geo-SCO_wf,
geo-sep-sym,
geo-sep_wf,
geo-between_wf,
geo-SCS_wf,
geo-point_wf,
geo-congruent_wf,
geo-colinear_wf,
set_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
equal_wf,
exists_wf,
not_wf,
squash_wf,
sq_stable__geo-congruent,
geo-between-symmetry,
sq_stable__geo-between,
sq_stable__and,
sq_stable__colinear,
sq_stable__all,
sq_stable__geo-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
functionEquality,
productElimination,
instantiate,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(a \mneq{} b
{}\mRightarrow{} c\_b\_d
{}\mRightarrow{} (\mexists{}u,v:Point. (cu \00D0 cd \mwedge{} cv \00D0 cd \mwedge{} a\_b\_u \mwedge{} (v\_b\_u \mwedge{} Colinear(a;b;v)) \mwedge{} (b \mneq{} d {}\mRightarrow{} v \mneq{} u))))
Date html generated:
2017_10_02-PM-04_40_51
Last ObjectModification:
2017_08_12-PM-08_36_40
Theory : euclidean!plane!geometry
Home
Index