Nuprl Lemma : geo-zero-angle-congruence-subst
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (abc ≅a xyx ⇒ aba ≅a xyx)
Proof
Definitions occuring in Statement :
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-out: out(p ab),
and: P ∧ Q,
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-zero-angle-congruence-out,
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-out_inversion,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
sqequalRule,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
productElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (abc \mcong{}\msuba{} xyx {}\mRightarrow{} aba \mcong{}\msuba{} xyx)
Date html generated:
2019_10_16-PM-01_28_09
Last ObjectModification:
2018_11_08-PM-00_22_05
Theory : euclidean!plane!geometry
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