Nuprl Lemma : geo-lt-angle_functionality
∀e:EuclideanPlane. ∀a,a',b,b',c,c',x,x',y,y',z,z':Point.
(a ≡ a' ⇒ b ≡ b' ⇒ c ≡ c' ⇒ x ≡ x' ⇒ y ≡ y' ⇒ z ≡ z' ⇒ (abc < xyz ⇐⇒ a'b'c' < x'y'z'))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-lt-angle: abc < xyz,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
not: ¬A,
false: False,
basic-geometry: BasicGeometry,
cand: A c∧ B
Lemmas referenced :
geo-lt-angle_wf,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-out_wf,
geo-between_wf,
geo-cong-angle_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-out_functionality,
geo-cong-angle_wf,
istype-void,
geo-sep_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
independent_functionElimination,
voidElimination,
dependent_pairFormation_alt,
productIsType,
functionIsType,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,a',b,b',c,c',x,x',y,y',z,z':Point.
(a \mequiv{} a' {}\mRightarrow{} b \mequiv{} b' {}\mRightarrow{} c \mequiv{} c' {}\mRightarrow{} x \mequiv{} x' {}\mRightarrow{} y \mequiv{} y' {}\mRightarrow{} z \mequiv{} z' {}\mRightarrow{} (abc < xyz \mLeftarrow{}{}\mRightarrow{} a'b'c' < x'y'z'))
Date html generated:
2019_10_16-PM-02_01_53
Last ObjectModification:
2019_09_27-PM-07_29_03
Theory : euclidean!plane!geometry
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