Nuprl Lemma : adjacent-right-angles-supplementary-using-geom-tactic
∀e:EuclideanPlane. ∀a,b,c,d:Point. (c-b-d ⇒ a ≠ b ⇒ (Rabc ⇐⇒ abc ≅a abd))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
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,
member: t ∈ T,
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,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
cand: A c∧ B,
geo-cong-angle: abc ≅a xyz,
right-angle: Rabc,
geo-midpoint: a=m=b,
uiff: uiff(P;Q)
Lemmas referenced :
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
geo-sep_wf,
geo-strict-between_wf,
geo-point_wf,
symmetric-point-construction,
geo-sep-sym,
geo-strict-between-sep2,
geo-strict-between-sym,
geo-strict-between-implies-colinear,
geo-colinear_wf,
right-angle-colinear2,
geo-colinear-symmetry,
right-angle-symmetry,
geo-right-angles-congruent,
geo-cong-angle-symmetry,
geo-congruent-flip,
geo-congruent_wf,
geo-congruent-refl,
geo-inner-five-segment,
geo-between_wf,
geo-between-symmetry,
geo-congruent-full-symmetry,
geo-construction-unicity,
geo-eq_wf,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-between-sep,
geo-between-implies-colinear,
right-angle-colinear,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-between-exchange4,
geo-congruent-iff-length,
geo-length-flip,
symmetric-point-unicity,
geo-midpoint_wf,
geo-midpoint-symmetry,
geo-midpoint_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
productElimination,
rename,
isectIsType,
functionIsType,
productIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (c-b-d {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} (Rabc \mLeftarrow{}{}\mRightarrow{} abc \mcong{}\msuba{} abd))
Date html generated:
2019_10_16-PM-01_55_09
Last ObjectModification:
2018_11_07-PM-01_03_07
Theory : euclidean!plane!geometry
Home
Index