Nuprl Lemma : r-triangle-inequality-rsub
∀[x,y:ℝ]. (|x - y| ≤ (|x| + |y|))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
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,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
r-triangle-inequality2,
int-to-real_wf,
less_than'_wf,
rsub_wf,
radd_wf,
rabs_wf,
real_wf,
nat_plus_wf,
rminus_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
radd_functionality,
rabs_functionality,
itermConstant_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rleq_wf,
squash_wf,
true_wf,
rabs-rminus,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
voidEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. (|x - y| \mleq{} (|x| + |y|))
Date html generated:
2017_10_03-AM-08_29_37
Last ObjectModification:
2017_07_28-AM-07_26_02
Theory : reals
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