Nuprl Lemma : real-vec-dist-identity
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(d(x;y) = r0;req-vec(n;x;y))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec: ℝ^n,
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
iff: P ⇐⇒ Q,
dot-product: x⋅y,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
guard: {T},
cand: A c∧ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B,
less_than: a < b,
so_apply: x[s],
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rge: x ≥ y,
real-vec-sub: X - Y,
rsub: x - y,
real-vec-mul: a*X,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
int_seg_wf,
req_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
rsqrt-is-zero,
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
equal_wf,
not-rneq,
rneq_wf,
rmul-is-positive,
rless_wf,
rsum-split,
subtract_wf,
rmul_wf,
subtract-add-cancel,
nat_plus_properties,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
rsum_wf,
radd_wf,
itermAdd_wf,
int_term_value_add_lemma,
req_functionality,
req_weakening,
rsum_nonneg,
square-nonneg,
le_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rsum-split-last,
rless_functionality,
radd_functionality,
trivial-rless-radd,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
radd-preserves-req,
rsub_wf,
rminus_wf,
uiff_transitivity,
radd-ac,
radd_comm,
radd-rminus-both,
radd-zero-both,
real-vec-norm_functionality,
real-vec-mul_wf,
rsub_functionality,
rmul_functionality,
rmul-zero-both,
real-vec-norm_wf,
rabs_wf,
req_transitivity,
real-vec-norm-mul,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
independent_functionElimination,
hypothesis,
natural_numberEquality,
setElimination,
rename,
setEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
lambdaFormation,
independent_isectElimination,
unionElimination,
inrFormation,
productEquality,
inlFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. uiff(d(x;y) = r0;req-vec(n;x;y))
Date html generated:
2017_10_03-AM-10_55_45
Last ObjectModification:
2017_07_28-AM-08_21_14
Theory : reals
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