Nuprl Lemma : vec-midpoint-dist
∀[n:ℕ]. ∀[a,b:ℝ^n]. (d(a;vec-midpoint(a;b)) = ((r1/r(2)) * d(a;b)))
Proof
Definitions occuring in Statement :
vec-midpoint: vec-midpoint(a;b),
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rdiv: (x/y),
req: x = y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
uiff: uiff(P;Q),
le: A ≤ B,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
real-vec-mul: a*X,
real-vec-sub: X - Y,
vec-midpoint: vec-midpoint(a;b),
req-vec: req-vec(n;x;y),
real-vec-add: X + Y,
int_seg: {i..j-},
lelt: i ≤ j < k,
real-vec: ℝ^n,
rat_term_to_real: rat_term_to_real(f;t),
rtermSubtract: left "-" right,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
rtermAdd: left "+" right,
pi2: snd(t)
Lemmas referenced :
rabs-of-nonneg,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rleq-int-fractions2,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
req_witness,
real-vec-dist_wf,
vec-midpoint_wf,
rmul_wf,
real-vec_wf,
istype-nat,
rabs_wf,
real-vec-mul_wf,
real-vec-dist-equal-iff,
req_functionality,
req_weakening,
rmul_functionality,
req_inversion,
real-vec-dist-dilation,
dot-product_wf,
real-vec-sub_wf,
dot-product-comm,
dot-product_functionality,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermVar_wf,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermAdd_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
universeIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
lambdaFormation_alt,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
int_eqEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbR{}\^{}n]. (d(a;vec-midpoint(a;b)) = ((r1/r(2)) * d(a;b)))
Date html generated:
2019_10_30-AM-11_32_26
Last ObjectModification:
2019_04_02-PM-04_09_01
Theory : reals!model!euclidean!geometry
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