Nuprl Lemma : r-triangle-inequality
∀[x,y:ℝ]. (|x + y| ≤ (|x| + |y|))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rleq: x ≤ y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rsub: x - y,
rabs: |x|,
nat: ℕ,
reg-seq-add: reg-seq-add(x;y),
rminus: -(x),
rnonneg2: rnonneg2(x),
exists: ∃x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
int_upper: {i...},
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j
Lemmas referenced :
int-triangle-inequality,
minus_functionality_wrt_le,
multiply_functionality_wrt_le,
le_weakening,
le_functionality,
int_term_value_mul_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
intformand_wf,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermMinus_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
int_upper_properties,
nat_plus_subtype_nat,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
decidable__lt,
less_than_transitivity1,
le_wf,
all_wf,
int_upper_wf,
less_than_wf,
bdd-diff_weakening,
reg-seq-add_functionality_wrt_bdd-diff,
reg-seq-add_wf,
rabs_functionality_wrt_bdd-diff,
nat_wf,
absval_wf,
rminus_functionality_wrt_bdd-diff,
radd-bdd-diff,
rminus_wf,
rnonneg2_functionality,
nat_plus_wf,
real_wf,
less_than'_wf,
rabs_wf,
radd_wf,
rsub_wf,
rnonneg-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
addEquality,
lambdaFormation,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
multiplyEquality,
independent_isectElimination,
unionElimination,
voidEquality,
intEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}[x,y:\mBbbR{}]. (|x + y| \mleq{} (|x| + |y|))
Date html generated:
2016_05_18-AM-07_14_23
Last ObjectModification:
2016_01_17-AM-01_55_03
Theory : reals
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