Nuprl Lemma : straight-angle-sum2_symm
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k:Point. (abc + xyz ≅ ijk ⇒ out(b ac) ⇒ (x-y-z ⇐⇒ i-j-k))
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
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,
hp-angle-sum: abc + xyz ≅ def,
exists: ∃x:A. B[x],
geo-cong-angle: abc ≅a xyz,
cand: A c∧ B,
member: t ∈ T,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-out: out(p ab),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
not: ¬A,
false: False,
stable: Stable{P},
geo-eq: a ≡ b
Lemmas referenced :
geo-sep-sym,
angle-cong-preserves-zero-angle,
geo-cong-angle-symm2,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
hp-angle-sum_wf,
geo-point_wf,
angle-cong-preserves-straight-angle,
geo-between-symmetry,
geo-strict-between-implies-between,
extended-out-preserves-between,
geo-out_inversion,
geo-strict-between-sym,
geo-between-inner-trans,
stable__geo-between,
false_wf,
geo-sep_wf,
not_wf,
geo-between_wf,
istype-void,
minimal-double-negation-hyp-elim,
geo-out_functionality,
geo-eq_weakening,
geo-strict-between_functionality,
geo-between_functionality,
geo-cong-angle_functionality,
minimal-not-not-excluded-middle,
geo-between-middle-or,
geo-between-exchange3,
geo-between-exchange4,
geo-between-outer-trans,
geo-out-iff-between1,
geo-between-outer-trans-cpy,
geo-not-bet-and-out
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
setElimination,
rename,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k:Point. (abc + xyz \mcong{} ijk {}\mRightarrow{} out(b ac) {}\mRightarrow{} (x-y-z \mLeftarrow{}{}\mRightarrow{} i-j-k))
Date html generated:
2019_10_16-PM-02_05_21
Last ObjectModification:
2019_06_24-AM-10_32_07
Theory : euclidean!plane!geometry
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