Nuprl Lemma : zero-angle-less-than-all
∀g:EuclideanPlane. ∀a,b,x,y,z:Point. (a ≠ b ⇒ x # yz ⇒ aba < xyz)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-sep: a ≠ b,
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,
geo-lt-angle: abc < xyz,
cand: A c∧ B,
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry,
not: ¬A,
false: False,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf,
geo-sep-sym,
lsep-implies-sep,
geo-between-trivial,
zero-angles-congruent,
not-out-if-lsep,
geo-out_weakening,
geo-eq_weakening,
geo-out-iff-colinear,
geo-between-trivial2,
euclidean-plane-axioms,
geo-between_wf,
geo-out_wf,
istype-void,
geo-cong-angle_wf
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,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
productIsType,
functionIsType,
dependent_pairFormation_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,x,y,z:Point. (a \mneq{} b {}\mRightarrow{} x \# yz {}\mRightarrow{} aba < xyz)
Date html generated:
2019_10_16-PM-01_58_14
Last ObjectModification:
2019_09_27-PM-05_56_10
Theory : euclidean!plane!geometry
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