Nuprl Lemma : hp-angle-sum-eq-straight-angle
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k,i',j',k':Point. (abc + xyz ≅ ijk ⇒ abc + xyz ≅ i'j'k' ⇒ i-j-k ⇒ i'-j'-k')
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
geo-out: out(p ab),
geo-strict-between: a-b-c,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
not: ¬A,
false: False,
stable: Stable{P},
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry,
geo-cong-angle: abc ≅a xyz,
basic-geometry-: BasicGeometry-,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
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,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
geo-eq: a ≡ b,
rev_implies: P ⇐ Q
Lemmas referenced :
euclidean-plane-axioms,
geo-sep-sym,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
hp-angle-sum_wf,
geo-point_wf,
stable__geo-between,
false_wf,
geo-lsep_wf,
not_wf,
geo-between_wf,
istype-void,
minimal-double-negation-hyp-elim,
not-lsep-iff-colinear,
minimal-not-not-excluded-middle,
cong-angle-out-exists-cong3,
geo-cong-angle-transitivity,
geo-cong-angle-symm2,
geo-proper-extend-exists,
geo-five-segment,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-congruent-iff-length,
geo-length-flip,
cong-angle-preserves-lsep_strong,
geo-congruent-sep,
lsep-implies-sep,
geo-strict-between-sep3,
cong-tri-implies-cong-angle2,
geo-sep_wf,
geo-out-iff-between1,
geo-out_inversion,
colinear-lsep,
lsep-all-sym,
geo-colinear-is-colinear-set,
geo-out-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,
left-symmetry,
Euclid-Prop7,
geo-left_wf,
geo-out_weakening,
geo-eq_weakening,
out-cong-angle,
geo-left-out-2,
geo-congruent-refl,
geo-sas2,
out-preserves-angle-cong_1,
extended-out-preserves-between,
geo-strict-between_functionality,
geo-left_functionality,
geo-out_functionality,
geo-congruent_functionality,
geo-sep_functionality,
geo-cong-angle_functionality,
geo-between_functionality,
not-left-and-right,
left-convex,
geo-left-out-1,
geo-between-out,
geo-strict-between-sep1,
between-preserves-left-2,
geo-strict-between-sep2,
left-between-implies-right1,
geo-out_transitivity,
geo-strict-between-sym,
geo-between-inner-trans,
geo-left-out-3,
left-between-implies-right2,
left-all-symmetry,
geo-colinear-cases,
geo-eq_wf,
straight-angle-sum2,
geo-between-trivial,
straight-angle-sum1_symm,
straight-angle-sum2_symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
setElimination,
rename,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
productIsType,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
promote_hyp,
inrFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k,i',j',k':Point.
(abc + xyz \mcong{} ijk {}\mRightarrow{} abc + xyz \mcong{} i'j'k' {}\mRightarrow{} i-j-k {}\mRightarrow{} i'-j'-k')
Date html generated:
2019_10_16-PM-02_25_30
Last ObjectModification:
2019_07_08-AM-11_11_22
Theory : euclidean!plane!geometry
Home
Index