Nuprl Lemma : rv-norm-triangle-inequality
∀[rv:InnerProductSpace]. ∀[x,y:Point(rv)]. (||x + y|| ≤ (||x|| + ||y||))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
rv-add: x + y,
rleq: x ≤ y,
radd: a + b,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
guard: {T},
nat: ℕ,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
itermConstant: "const"
Lemmas referenced :
rnexp-rleq-iff,
rv-norm_wf,
rv-add_wf,
inner-product-space_subtype,
radd_wf,
rv-norm-nonneg,
radd-non-neg,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
le_witness_for_triv,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rnexp_wf,
istype-void,
istype-le,
rv-ip_wf,
rmul_wf,
int-to-real_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
rleq_functionality,
rv-norm-squared,
req_weakening,
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,
radd_functionality,
req_inversion,
radd-preserves-rleq,
rminus_wf,
itermMinus_wf,
req_transitivity,
rv-ip-add,
rv-ip-add2,
real_term_value_minus_lemma,
rv-ip-symmetry,
rv-ip-rleq,
rleq-implies-rleq,
real_term_polynomial,
nat_plus_wf,
rsub_wf,
less_than'_wf,
false_wf,
rleq-int,
req_wf,
rleq_wf,
real_wf,
rmul_preserves_rleq2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
Error :memTop,
universeIsType,
voidElimination,
productElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
independent_pairFormation,
lambdaFormation_alt,
int_eqEquality,
voidEquality,
isect_memberEquality,
intEquality,
computeAll,
axiomEquality,
minusEquality,
independent_pairEquality,
lambdaFormation,
productEquality,
setEquality,
lambdaEquality,
isect_memberFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y:Point(rv)]. (||x + y|| \mleq{} (||x|| + ||y||))
Date html generated:
2020_05_20-PM-01_11_48
Last ObjectModification:
2019_12_09-PM-11_44_22
Theory : inner!product!spaces
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