Nuprl Lemma : geo-cong-angle-preserves-lt-angle
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z:Point. (abc ≅a def ⇒ abc < xyz ⇒ def < xyz)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
not: ¬A,
false: False,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-out_wf,
geo-lt-angle_wf,
geo-cong-angle_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle-transitivity,
euclidean-plane-axioms,
geo-cong-angle-symm2,
geo-between_wf,
geo-sep_wf,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
independent_pairFormation,
cut,
thin,
productElimination,
hypothesis,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
hypothesisEquality,
dependent_functionElimination,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
dependent_pairFormation_alt,
because_Cache,
productIsType,
functionIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z:Point. (abc \mcong{}\msuba{} def {}\mRightarrow{} abc < xyz {}\mRightarrow{} def < xyz)
Date html generated:
2019_10_16-PM-01_59_54
Last ObjectModification:
2019_10_02-AM-10_31_18
Theory : euclidean!plane!geometry
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