Nuprl Lemma : p6-aux-bet-preserves-angle
∀e:HeytingGeometry. ∀a,b,c,d:Point. (a # bc ⇒ (a_d_b ∧ a ≠ d) ⇒ cab ≅a cad)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-angle: abc ≅a xyz,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-cong-angle: abc ≅a xyz,
member: t ∈ T,
heyting-geometry: HeytingGeometry,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
extend-using-SC,
geo-triangle-property,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-triangle_wf,
geo-point_wf,
geo-between-trivial,
geo-congruent-refl,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
because_Cache,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
dependent_pairFormation_alt
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,d:Point. (a \# bc {}\mRightarrow{} (a\_d\_b \mwedge{} a \mneq{} d) {}\mRightarrow{} cab \mcong{}\msuba{} cad)
Date html generated:
2019_10_16-PM-02_10_03
Last ObjectModification:
2018_11_07-PM-01_09_44
Theory : euclidean!plane!geometry
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