Nuprl Lemma : geo-cong-angle-preserves-lt-angle2
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z:Point. (def ≅a xyz ⇒ abc < xyz ⇒ d # ef ⇒ abc < def)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-cong-angle: abc ≅a xyz,
false: False,
not: ¬A,
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
geo-cong-angle_wf,
geo-lt-angle_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-lsep_wf,
interior-point-cong-angle-transfer-full,
not-out-if-lsep,
geo-out_inversion,
geo-sep-sym,
geo-eq_weakening,
geo-out_weakening,
geo-cong-angle-symm2,
istype-void,
geo-sep_wf,
geo-between_wf,
geo-out_wf,
out-preserves-angle-cong_1
Rules used in proof :
because_Cache,
inhabitedIsType,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
universeIsType,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
functionIsType,
productIsType,
dependent_pairFormation_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z:Point. (def \mcong{}\msuba{} xyz {}\mRightarrow{} abc < xyz {}\mRightarrow{} d \# ef {}\mRightarrow{} abc < def)
Date html generated:
2019_10_16-PM-01_51_36
Last ObjectModification:
2019_10_02-PM-01_15_54
Theory : euclidean!plane!geometry
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