Nuprl Lemma : adjacent-right-angles-supplementary
∀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],
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
cand: A c∧ B,
right-angle: Rabc,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
geo-cong-angle: abc ≅a xyz,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m
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-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep2,
geo-strict-between-sep3,
geo-cong-angle-transitivity,
geo-midpoint_wf,
geo-congruent-refl,
geo-congruent-iff-length,
geo-length-flip,
geo-right-angles-congruent,
geo-five-segment',
vertical-angles-congruent,
geo-strict-between-sym,
geo-between-symmetry,
geo-construction-unicity,
geo-inner-three-segment,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-between-exchange4,
symmetric-point-unicity,
geo-midpoint-symmetry,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
right-angle-colinear,
geo-between-sep,
right-angle-symmetry
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,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType
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_53_01
Last ObjectModification:
2019_08_20-PM-01_36_40
Theory : euclidean!plane!geometry
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