Nuprl Lemma : Euclid-Prop26-2
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(a # bc ⇒ x # yz ⇒ abc ≅a xyz ⇒ bac ≅a yxz ⇒ ac ≅ xz ⇒ (ab ≅ xy ∧ bc ≅ yz ∧ bca ≅a yzx))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
or: P ∨ Q,
false: False,
not: ¬A,
geo-gt: cd > ab,
squash: ↓T,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
geo-out: out(p ab),
geo-tri: Triangle(a;b;c),
uiff: uiff(P;Q)
Lemmas referenced :
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
geo-lsep_wf,
geo-point_wf,
geo-ge-cases,
stable__geo-congruent,
double-negation-hyp-elim,
geo-gt_wf,
not_wf,
istype-void,
geo-congruent-sep,
lsep-implies-sep,
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
geo-strict-between-sep2,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
Euclid-prop16,
colinear-lsep-cycle,
geo-strict-between-sym,
lsep-symmetry,
geo-between-out,
geo-between-symmetry,
geo-out_weakening,
geo-eq_weakening,
geo-lt-angle-symm,
geo-out-preserves-lt-angle,
lt-angle-not-cong2,
geo-sas2,
p4geo,
geo-between_wf,
out-preserves-angle-cong_1,
euclidean-plane-axioms,
geo-cong-angle-symm2,
geo-cong-angle-transitivity,
geo-congruent-iff-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
independent_pairFormation,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
setElimination,
rename,
unionEquality,
independent_functionElimination,
unionElimination,
voidElimination,
lambdaEquality_alt,
functionIsType,
unionIsType,
imageElimination,
productElimination,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
productIsType,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(a \# bc {}\mRightarrow{} x \# yz {}\mRightarrow{} abc \mcong{}\msuba{} xyz {}\mRightarrow{} bac \mcong{}\msuba{} yxz {}\mRightarrow{} ac \mcong{} xz {}\mRightarrow{} (ab \mcong{} xy \mwedge{} bc \mcong{} yz \mwedge{} bca \mcong{}\msuba{} yzx))
Date html generated:
2019_10_16-PM-02_35_56
Last ObjectModification:
2019_09_12-AM-11_58_07
Theory : euclidean!plane!geometry
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