Nuprl Lemma : isosc-bisectors-between-ns
∀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-congruent: ab ≅ cd,
geo-between: a_b_c,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
geo-strict-between: a-b-c,
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
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,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-between_wf,
geo-triangle_wf,
geo-point_wf,
stable__geo-between,
false_wf,
or_wf,
geo-sep_wf,
not_wf,
minimal-double-negation-hyp-elim,
geo-congruent_functionality,
geo-eq_weakening,
geo-midpoint_functionality,
geo-between_functionality,
minimal-not-not-excluded-middle,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-congruent-sep,
geo-congruent-symmetry,
geo-sep-sym,
geo-triangle-property,
isosc-bisectors-between,
geo-congruence-identity3,
geo-eq_inversion,
geo-congruence-identity,
geo-triangle_functionality,
at-most-one-midpoint,
geo-between-trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
setEquality,
productEquality,
cumulativity,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
functionEquality,
independent_functionElimination,
unionElimination,
voidElimination,
independent_pairFormation,
promote_hyp
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_43
Last ObjectModification:
2017_08_16-AM-11_13_24
Theory : euclidean!plane!geometry
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