Nuprl Lemma : congruent-half-plane-angles-implies-right-angles
∀g:EuclideanPlane. ∀a,b,c,d:Point. (c-b-d ⇒ a # cd ⇒ abc ≅a abd ⇒ {Rabd ∧ Rabc})
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
right-angle: Rabc,
geo-strict-between: a-b-c,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
oriented-plane: OrientedPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
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,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
basic-geometry-: BasicGeometry-
Lemmas referenced :
lsep-colinear-sep,
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,
adjacent-right-angles-supplementary,
geo-cong-angle_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-point_wf,
geo-cong-angle-symm2,
geo-strict-between-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
isectElimination,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType,
productElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (c-b-d {}\mRightarrow{} a \# cd {}\mRightarrow{} abc \mcong{}\msuba{} abd {}\mRightarrow{} \{Rabd \mwedge{} Rabc\})
Date html generated:
2019_10_16-PM-01_55_22
Last ObjectModification:
2018_11_07-PM-01_03_28
Theory : euclidean!plane!geometry
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