Nuprl Lemma : rabs-int-rmul-unit
∀[k:ℕ]. ∀[x:ℝ]. (|-1^k * x| = |x|)
Proof
Definitions occuring in Statement :
rabs: |x|,
int-rmul: k1 * a,
req: x = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
fastexp: i^n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
absval: |i|
Lemmas referenced :
req_witness,
rabs_wf,
int-rmul_wf,
fastexp_wf,
real_wf,
nat_wf,
absval_wf,
req_functionality,
rabs-int-rmul,
req_weakening,
absval_exp,
exp-fastexp,
iff_weakening_equal,
exp-one,
true_wf,
squash_wf,
equal_wf,
exp-positive-stronger,
int_subtype_base,
less_than_wf,
set_subtype_base,
nat_plus_wf,
subtype_base_sq,
int-rmul-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
intEquality,
cumulativity,
instantiate
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. (|-1\^{}k * x| = |x|)
Date html generated:
2017_10_03-AM-08_29_05
Last ObjectModification:
2017_07_28-AM-07_25_41
Theory : reals
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