Nuprl Lemma : right-angle-sum
∀g:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k:Point.
(a ≠ b ⇒ b ≠ c ⇒ x ≠ y ⇒ y ≠ z ⇒ Rabc ⇒ Rxyz ⇒ i-j-k ⇒ abc + xyz ≅ ijk)
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
guard: {T}
Lemmas referenced :
Euclid-erect-perp,
geo-strict-between-sep1,
geo-sep_wf,
geo-colinear-is-colinear-set,
geo-strict-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,
geo-colinear_wf,
sq_stable__and,
geo-perp-in_wf,
geo-lsep_wf,
sq_stable__geo-perp-in,
sq_stable__geo-lsep,
lsep-iff-all-sep,
geo-sep-sym,
geo-right-angles-congruent,
geo-colinear-same,
geo-strict-between-sep2,
geo-strict-between-sep3,
geo-between-trivial2,
geo-out_weakening,
geo-eq_weakening,
geo-cong-angle_wf,
geo-between_wf,
geo-out_wf,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
right-angle_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
because_Cache,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
applyEquality,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
setElimination,
rename,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType,
instantiate
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k:Point.
(a \mneq{} b {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} x \mneq{} y {}\mRightarrow{} y \mneq{} z {}\mRightarrow{} Rabc {}\mRightarrow{} Rxyz {}\mRightarrow{} i-j-k {}\mRightarrow{} abc + xyz \mcong{} ijk)
Date html generated:
2019_10_16-PM-02_05_35
Last ObjectModification:
2019_06_05-AM-09_36_44
Theory : euclidean!plane!geometry
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