Nuprl Lemma : out-preserves-lsep
∀g:EuclideanPlane. ∀a,b,c,b',c':Point. (a # bc ⇒ out(a bb') ⇒ out(a cc') ⇒ a # b'c')
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
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,
uall: ∀[x:A]. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
colinear-lsep,
lsep-symmetry,
lsep-all-sym,
euclidean-plane-axioms,
geo-colinear-permute,
geo-colinear-is-colinear-set,
geo-out-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
colinear-lsep-cycle,
geo-sep-sym,
geo-out_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
productElimination,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
isectElimination,
universeIsType,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,b',c':Point. (a \# bc {}\mRightarrow{} out(a bb') {}\mRightarrow{} out(a cc') {}\mRightarrow{} a \# b'c')
Date html generated:
2019_10_16-PM-01_43_30
Last ObjectModification:
2018_10_23-PM-03_22_29
Theory : euclidean!plane!geometry
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