Nuprl Lemma : hp-angle-sum-eq3
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(abc ≅a a'b'c' ⇒ ijk ≅a i'j'k' ⇒ abc + xyz ≅ ijk ⇒ a'b'c' + x'y'z' ≅ i'j'k' ⇒ a # bc ⇒ i # jk ⇒ xyz ≅a x'y'z')
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
oriented-plane: OrientedPlane,
or: P ∨ Q,
geo-lsep: a # bc,
true: True,
squash: ↓T,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-tri: Triangle(a;b;c),
false: False,
not: ¬A,
uiff: uiff(P;Q),
geo-cong-angle: abc ≅a xyz,
basic-geometry-: BasicGeometry-,
cand: A c∧ B,
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
hp-angle-sum: abc + xyz ≅ def,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
between-preserves-left-3,
between-preserves-left-5,
geo-strict-between_functionality,
geo-out_functionality,
geo-congruent_functionality,
geo-left_functionality,
geo-cong-angle_functionality,
out-cong-angle,
geo-cong-angle-symm3,
geo-cong-angle-symmetry,
geo-eq_inversion,
geo-left_wf,
Euclid-Prop7,
between-preserves-left-1,
left-implies-sep,
geo-between-exchange4,
geo-between-exchange3,
between-preserves-left-2,
geo-left-out-4,
left-symmetry,
geo-between-out,
basic-geometry_wf,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-add-length-between,
geo-congruent-refl,
geo-strict-between-sep2,
p8geo,
geo-between-trivial,
geo-length-flip,
geo-congruent_wf,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep1,
geo-cong-angle-transitivity,
geo-eq_weakening,
geo-out_weakening,
geo-out_inversion,
istype-void,
geo-between_wf,
geo-between-sep,
lsep-implies-sep,
out-preserves-angle-cong_1,
geo-congruent-iff-length,
geo-sas2,
geo-out_transitivity,
euclidean-plane-axioms,
geo-out-interior-point-exists,
lsep-symmetry,
geo-cong-angle-symm2,
cong-angle-preserves-lsep_strong,
geo-strict-between-sym,
geo-out-if-between,
lsep-all-sym,
out-preserves-lsep,
geo-strict-between-sep3,
geo-sep-O-X,
geo-sep-sym,
geo-X_wf,
geo-O_wf,
geo-proper-extend-exists,
geo-point_wf,
geo-cong-angle_wf,
hp-angle-sum_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-lsep_wf
Rules used in proof :
promote_hyp,
dependent_set_memberEquality_alt,
unionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality_alt,
dependent_pairFormation_alt,
functionIsType,
productIsType,
voidElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
rename,
setElimination,
productElimination,
inhabitedIsType,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
universeIsType,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(abc \mcong{}\msuba{} a'b'c'
{}\mRightarrow{} ijk \mcong{}\msuba{} i'j'k'
{}\mRightarrow{} abc + xyz \mcong{} ijk
{}\mRightarrow{} a'b'c' + x'y'z' \mcong{} i'j'k'
{}\mRightarrow{} a \# bc
{}\mRightarrow{} i \# jk
{}\mRightarrow{} xyz \mcong{}\msuba{} x'y'z')
Date html generated:
2019_10_29-AM-09_21_19
Last ObjectModification:
2019_10_18-PM-04_51_08
Theory : euclidean!plane!geometry
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