Nuprl Lemma : Prop22-symmetric-point-construction-lemma
∀e:EuclideanPlane. ∀a,b,c:Point.
(a # bc
⇒ (∃c1,c1',c1'',c2,c2',c2'':Point
((ac1 ≅ ac ∧ out(a bc1))
∧ (bc2 ≅ bc ∧ out(b ac2))
∧ (b-a-c1' ∧ ac1' ≅ ac1)
∧ (a-b-c2' ∧ bc2' ≅ bc2)
∧ (a_b_c1'' ∧ bc1'' ≅ bc1)
∧ (b_a_c2'' ∧ ac2'' ≅ ac2)
∧ (a-c1-c2' ∧ b-c2-c1')
∧ (c1'-a-c1 ∧ c2'-b-c2)
∧ (c1'-c2-c2' ∧ c2'-c1-c1')
∧ c1'-c2-c1
∧ c2'-c1-c2)))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
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,
member: t ∈ T,
and: P ∧ Q,
euclidean-plane: EuclideanPlane,
guard: {T},
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-out: out(p ab),
geo-strict-between: a-b-c
Lemmas referenced :
Euclid-Prop20_cycle,
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-sym,
lsep-implies-sep,
geo-sep-O-X,
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,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-length-flip,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
subtype_rel_self,
iff_weakening_equal,
geo-add-length_functionality_wrt_cong,
geo-add-length-comm,
geo-strict-between-sep3,
geo-triangle-inequality-lt-sep,
extend-using-SC,
geo-out_wf,
geo-between_wf,
geo-congruent_wf,
geo-strict-between_wf,
geo-out-iff-between1,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-congruent-iff-length,
geo-add-length-between,
equal_wf,
istype-universe,
geo-lt-out-to-between,
geo-strict-between-sep1,
extended-out-preserves-between,
geo-between-exchange4,
geo-out_inversion,
geo-between-implies-out2,
geo-strict-between-sym,
geo-length_wf1,
geo-add-length_wf1,
geo-zero-point-sep-iff-sep,
geo-lt-sep,
geo-length-equality,
geo-lt-add1-iff,
geo-between-inner-trans,
geo-between-out,
geo-strict-between-sep2,
euclidean-plane-axioms,
geo-out_transitivity,
geo-lt-add1-iff2,
geo-add-length-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
because_Cache,
setElimination,
rename,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairFormation,
dependent_pairFormation_alt,
productIsType,
dependent_set_memberEquality_alt,
equalityIstype,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
(a \# bc
{}\mRightarrow{} (\mexists{}c1,c1',c1'',c2,c2',c2'':Point
((ac1 \mcong{} ac \mwedge{} out(a bc1))
\mwedge{} (bc2 \mcong{} bc \mwedge{} out(b ac2))
\mwedge{} (b-a-c1' \mwedge{} ac1' \mcong{} ac1)
\mwedge{} (a-b-c2' \mwedge{} bc2' \mcong{} bc2)
\mwedge{} (a\_b\_c1'' \mwedge{} bc1'' \mcong{} bc1)
\mwedge{} (b\_a\_c2'' \mwedge{} ac2'' \mcong{} ac2)
\mwedge{} (a-c1-c2' \mwedge{} b-c2-c1')
\mwedge{} (c1'-a-c1 \mwedge{} c2'-b-c2)
\mwedge{} (c1'-c2-c2' \mwedge{} c2'-c1-c1')
\mwedge{} c1'-c2-c1
\mwedge{} c2'-c1-c2)))
Date html generated:
2019_10_16-PM-02_22_25
Last ObjectModification:
2019_03_06-AM-11_30_17
Theory : euclidean!plane!geometry
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