Nuprl Lemma : isosc-bisectors-between
∀e:HeytingGeometry. ∀a,b,c,m,a',b',m':Point.
(c # ab ⇒ ac ≅ bc ⇒ (c-a-a' ∧ b'-b-c) ⇒ a=m=b ⇒ a'=m'=b' ⇒ aa' ≅ bb' ⇒ c-m-m')
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
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,
member: t ∈ T,
guard: {T},
cand: A c∧ B,
heyting-geometry: HeytingGeometry,
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
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,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-triangle-colinear,
geo-triangle-symmetry,
geo-sep-sym,
geo-strict-between-sep1,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
subtype_rel_self,
geo-triangle-property,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
midpoint-sep,
geo-between-implies-colinear,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-midpoint_wf,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-strict-between_wf,
geo-triangle_wf,
geo-point_wf,
geo-add-length-between,
geo-congruent-iff-length,
geo-length-flip,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
isosceles-mid-exists,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-krippen-lemma,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-exchange3,
geo-between-exchange4,
geo-between-inner-trans,
geo-congruent-symmetry,
geo-midpoint-symmetry,
geo-out-iff-between1,
geo-out-colinear,
geo-strict-between-sym,
double-pasch-exists,
geo-intersection-unicity,
geo-colinear_wf,
not-geo-triangle-iff-colinear,
geo-eq_weakening,
geo-strict-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
setEquality,
productEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
rename
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,m,a',b',m':Point.
(c \# ab {}\mRightarrow{} ac \00D0 bc {}\mRightarrow{} (c-a-a' \mwedge{} b'-b-c) {}\mRightarrow{} a=m=b {}\mRightarrow{} a'=m'=b' {}\mRightarrow{} aa' \00D0 bb' {}\mRightarrow{} c-m-m')
Date html generated:
2017_10_02-PM-07_06_23
Last ObjectModification:
2017_08_16-PM-00_17_13
Theory : euclidean!plane!geometry
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