Nuprl Lemma : vert-angles-congruent
∀e:HeytingGeometry. ∀b,a,a',c,c':Point. (a-b-a' ⇒ c-b-c' ⇒ a # bc ⇒ abc ≅a a'bc')
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-angle: abc ≅a xyz,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
heyting-geometry: HeytingGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-triangle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-point_wf,
geo-triangle-symmetry,
sup-angles-preserve-congruence,
geo-cong-angle-symm,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-strict-between-sep2,
geo-triangle-colinear,
geo-sep-sym,
geo-strict-between-sep3,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-strict-between-sym,
subtype_rel_self,
basic-geometry-_wf,
euclidean-plane-axioms,
geo-cong-angle-transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
productElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType
Latex:
\mforall{}e:HeytingGeometry. \mforall{}b,a,a',c,c':Point. (a-b-a' {}\mRightarrow{} c-b-c' {}\mRightarrow{} a \# bc {}\mRightarrow{} abc \mcong{}\msuba{} a'bc')
Date html generated:
2019_10_16-PM-02_09_03
Last ObjectModification:
2018_11_07-PM-01_08_50
Theory : euclidean!plane!geometry
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