Nuprl Lemma : rn-metric-leq-rn-prod-metric
∀[n:ℕ]. rn-metric(n) ≤ rn-prod-metric(n)
Proof
Definitions occuring in Statement : 
rn-prod-metric: rn-prod-metric(n)
, 
rn-metric: rn-metric(n)
, 
real-vec: ℝ^n
, 
metric-leq: d1 ≤ d2
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
Definitions unfolded in proof : 
rn-metric: rn-metric(n)
, 
metric-leq: d1 ≤ d2
, 
mdist: mdist(d;x;y)
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
real-vec-dist: d(x;y)
, 
rleq: x ≤ y
, 
rnonneg: rnonneg(x)
, 
le: A ≤ B
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
nat: ℕ
, 
less_than': less_than'(a;b)
, 
not: ¬A
, 
false: False
, 
rn-prod-metric: rn-prod-metric(n)
, 
rmetric: rmetric()
, 
prod-metric: prod-metric(k;d)
, 
real-vec-sub: X - Y
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
ge: i ≥ j 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
dot-product: x⋅y
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
less_than: a < b
, 
squash: ↓T
, 
subtract: n - m
, 
true: True
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
real-vec: ℝ^n
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
nat_plus: ℕ+
, 
rge: x ≥ y
, 
guard: {T}
, 
req_int_terms: t1 ≡ t2
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m]
Lemmas referenced : 
square-rleq-implies, 
real-vec-dist_wf, 
mdist_wf, 
real-vec_wf, 
rn-prod-metric_wf, 
mdist-nonneg, 
le_witness_for_triv, 
istype-nat, 
rnexp_wf, 
istype-void, 
istype-le, 
real-vec-norm_wf, 
real-vec-sub_wf, 
dot-product_wf, 
rleq_functionality, 
real-vec-norm-squared, 
req_weakening, 
nat_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
ge_wf, 
istype-less_than, 
subtract-1-ge-0, 
decidable__le, 
intformnot_wf, 
int_formula_prop_not_lemma, 
subtract_wf, 
rnexp2-nonneg, 
int-to-real_wf, 
rsum-empty, 
rsum_wf, 
rabs_wf, 
int_seg_properties, 
decidable__lt, 
itermAdd_wf, 
itermSubtract_wf, 
int_term_value_add_lemma, 
int_term_value_subtract_lemma, 
int_seg_wf, 
radd_wf, 
real-vec-subtype, 
rmul_wf, 
itermMultiply_wf, 
radd-preserves-rleq, 
rminus_wf, 
itermMinus_wf, 
rnexp_functionality, 
rsum-split-last, 
dot-product-split-last, 
rleq_functionality_wrt_implies, 
radd_functionality_wrt_rleq, 
rleq_weakening_equal, 
radd_functionality, 
rnexp2, 
req-iff-rsub-is-0, 
real_polynomial_null, 
real_term_value_sub_lemma, 
real_term_value_mul_lemma, 
real_term_value_add_lemma, 
real_term_value_var_lemma, 
real_term_value_const_lemma, 
req_inversion, 
rabs-rnexp2, 
real_term_value_minus_lemma, 
rleq_wf, 
squash_wf, 
true_wf, 
real_wf, 
subtype_rel_self, 
iff_weakening_equal, 
rsum_nonneg, 
zero-rleq-rabs, 
rmul_preserves_rleq2, 
rmul-nonneg-case1, 
rleq-int, 
istype-false
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation_alt, 
introduction, 
cut, 
lambdaFormation_alt, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
lambdaEquality_alt, 
setElimination, 
rename, 
inhabitedIsType, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
universeIsType, 
productElimination, 
independent_isectElimination, 
functionIsTypeImplies, 
dependent_set_memberEquality_alt, 
natural_numberEquality, 
independent_pairFormation, 
voidElimination, 
because_Cache, 
equalityIstype, 
intWeakElimination, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
unionElimination, 
minusEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
productIsType, 
addEquality, 
instantiate, 
universeEquality
Latex:
\mforall{}[n:\mBbbN{}].  rn-metric(n)  \mleq{}  rn-prod-metric(n)
Date html generated:
2019_10_30-AM-08_37_16
Last ObjectModification:
2019_10_02-AM-11_02_55
Theory : reals
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