Nuprl Lemma : meq-rn-prod-metric
∀[k:ℕ]. ∀[x,y:ℝ^k]. uiff(x ≡ y;req-vec(k;x;y))
Proof
Definitions occuring in Statement :
rn-prod-metric: rn-prod-metric(n),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
meq: x ≡ y,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
absval: |i|,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
implies: P ⇒ Q,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
req-vec: req-vec(n;x;y),
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x],
mdist: mdist(d;x;y),
meq: x ≡ y
Lemmas referenced :
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rabs-int,
req_transitivity,
req_weakening,
rsub_functionality,
rabs_functionality,
req_functionality,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
req-int,
rabs-difference-is-zero,
subtract-add-cancel,
istype-nat,
real-vec_wf,
zero-rleq-rabs,
rsum-of-nonneg-zero-iff,
rsum_wf,
iff_weakening_uiff,
int-to-real_wf,
istype-less_than,
istype-le,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
rsub_wf,
rabs_wf,
req_wf,
subtract_wf,
req_witness,
int_seg_wf,
istype-void,
mdist-rn-prod-metric
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
minusEquality,
isectIsTypeImplies,
independent_pairEquality,
promote_hyp,
functionEquality,
because_Cache,
productIsType,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
imageElimination,
productElimination,
dependent_set_memberEquality_alt,
addEquality,
functionIsType,
inhabitedIsType,
functionIsTypeImplies,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
lambdaEquality_alt,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
universeIsType,
lambdaFormation_alt,
isect_memberFormation_alt,
independent_pairFormation,
hypothesis,
voidElimination,
isect_memberEquality_alt,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}k]. uiff(x \mequiv{} y;req-vec(k;x;y))
Date html generated:
2019_10_30-AM-08_34_31
Last ObjectModification:
2019_10_27-PM-04_57_17
Theory : reals
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