Nuprl Lemma : cong-angle-between-exists-iff
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.
((((b ≠ a ∧ b ≠ c) ∧ y ≠ x) ∧ y ≠ z)
⇒ (abc ≅a xyz
⇐⇒ ∃a',c',x',z':Point
((((b_a_a' ∧ b_c_c') ∧ y_x_x') ∧ y_z_z') ∧ (((ba ≅ xx' ∧ aa' ≅ yx) ∧ bc ≅ zz') ∧ cc' ≅ yz) ∧ a'c' ≅ x'z')))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-cong-angle: abc ≅a xyz,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
or: P ∨ Q,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
cand: A c∧ B
Lemmas referenced :
geo-cong-angle_wf,
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,
geo-proper-extend-exists,
geo-length-flip,
geo-between-out,
out-congruent,
geo-add-length-comm,
geo-length-type_wf,
true_wf,
squash_wf,
geo-add-length_wf,
geo-add-length-between,
less_than_wf,
le_wf,
istype-false,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
istype-void,
list_ind_cons_lemma,
geo-strict-between-implies-colinear,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
oriented-plane_wf,
euclidean-plane-subtype-oriented,
oriented-colinear-append,
geo-between-sep,
geo-colinear-five-segment,
geo-congruent-iff-length,
basic-geometry-_wf,
subtype_rel_self,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-sep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
inhabitedIsType,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
equalityTransitivity,
imageElimination,
lambdaEquality_alt,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
equalityIsType1,
inrFormation_alt,
inlFormation_alt,
equalitySymmetry,
dependent_pairFormation_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
((((b \mneq{} a \mwedge{} b \mneq{} c) \mwedge{} y \mneq{} x) \mwedge{} y \mneq{} z)
{}\mRightarrow{} (abc \mcong{}\msuba{} xyz
\mLeftarrow{}{}\mRightarrow{} \mexists{}a',c',x',z':Point
((((b\_a\_a' \mwedge{} b\_c\_c') \mwedge{} y\_x\_x') \mwedge{} y\_z\_z')
\mwedge{} (((ba \mcong{} xx' \mwedge{} aa' \mcong{} yx) \mwedge{} bc \mcong{} zz') \mwedge{} cc' \mcong{} yz)
\mwedge{} a'c' \mcong{} x'z')))
Date html generated:
2019_10_16-PM-01_27_00
Last ObjectModification:
2018_12_11-PM-11_08_45
Theory : euclidean!plane!geometry
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