Nuprl Lemma : geo-congruent-preserves-out
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point. (bc ≅ b'c' ⇒ ac ≅ a'c' ⇒ ab ≅ a'b' ⇒ out(a bc) ⇒ out(a' b'c'))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
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,
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,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
uall: ∀[x:A]. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
geo-out: out(p ab),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
cand: A c∧ B,
geo-strict-between: a-b-c
Lemmas referenced :
geo-colinear-is-colinear-set,
geo-out-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-congruent-symmetry,
geo-congruent-sep,
geo-colinear-cases,
subtype_rel_self,
basic-geometry-_wf,
not_wf,
geo-between_wf,
stable__not,
geo-eq_wf,
geo-strict-between_wf,
geo-colinear_wf,
geo-out_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-congruence-identity,
geo-between-trivial,
geo-colinear_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-sep_functionality,
geo-congruent_functionality,
geo-sep-irrefl',
geo-congruent-preserves-between,
geo-strict-between-implies-between,
geo-between-out,
geo-congruent-iff-length,
geo-length-flip,
geo-between-symmetry,
euclidean-plane-axioms,
geo-sep-sym,
geo-out_inversion,
geo-not-bet-and-out
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isectElimination,
productElimination,
independent_isectElimination,
applyEquality,
instantiate,
productEquality,
comment,
addLevel,
impliesFunctionality,
andLevelFunctionality,
impliesLevelFunctionality,
levelHypothesis,
promote_hyp,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point.
(bc \mcong{} b'c' {}\mRightarrow{} ac \mcong{} a'c' {}\mRightarrow{} ab \mcong{} a'b' {}\mRightarrow{} out(a bc) {}\mRightarrow{} out(a' b'c'))
Date html generated:
2018_05_22-AM-11_58_45
Last ObjectModification:
2018_04_06-PM-00_06_16
Theory : euclidean!plane!geometry
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