Nuprl Lemma : geo-cong-angle_weakening
∀e:BasicGeometry. ∀a,b,c:Point. ((a ≠ b ∧ a ≡ c) ⇒ abc ≅a abc)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
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,
cand: A c∧ B,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-sep-sym,
geo-sep_functionality,
geo-eq_inversion,
geo-eq_weakening,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-eq_wf,
geo-point_wf,
geo-between-trivial,
geo-congruent-refl,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
hypothesisEquality,
isectElimination,
independent_isectElimination,
sqequalRule,
productIsType,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType,
dependent_pairFormation_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. ((a \mneq{} b \mwedge{} a \mequiv{} c) {}\mRightarrow{} abc \mcong{}\msuba{} abc)
Date html generated:
2019_10_16-PM-01_27_49
Last ObjectModification:
2018_11_07-PM-00_56_15
Theory : euclidean!plane!geometry
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