Nuprl Lemma : geo-congruent-between-exists2
∀e:BasicGeometry. ∀a,b,c,a',c':Point.
(a ≠ c ⇒ (∃b':Point. (a'_b'_c' ∧ ab ≅ a'b' ∧ bc ≅ b'c')) supposing (a_b_c and ac ≅ a'c'))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
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,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
basic-geometry: BasicGeometry,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-between_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-congruent_wf,
geo-sep_wf,
geo-point_wf,
euclidean-plane-axioms,
geo-congruent-symmetry,
geo-congruent-sep,
geo-sep-sym,
extend-using-SC,
geo-extend-exists,
geo-between-sep,
geo-construction-unicity,
subtype_rel_self,
basic-geometry-_wf,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
geo-three-segment,
geo-congruent-iff-length,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
rename,
equalityTransitivity,
equalitySymmetry,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',c':Point.
(a \mneq{} c {}\mRightarrow{} (\mexists{}b':Point. (a'\_b'\_c' \mwedge{} ab \mcong{} a'b' \mwedge{} bc \mcong{} b'c')) supposing (a\_b\_c and ac \mcong{} a'c'))
Date html generated:
2019_10_16-PM-01_16_53
Last ObjectModification:
2019_04_19-PM-03_33_00
Theory : euclidean!plane!geometry
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