Nuprl Lemma : scale-metric_wf
∀[X:Type]. ∀[c:{c:ℝ| r0 ≤ c} ]. ∀[d:metric(X)].  (c*d ∈ metric(X))
Proof
Definitions occuring in Statement : 
scale-metric: c*d
, 
metric: metric(X)
, 
rleq: x ≤ y
, 
int-to-real: r(n)
, 
real: ℝ
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
natural_number: $n
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
metric: metric(X)
, 
scale-metric: c*d
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
uimplies: b supposing a
, 
sq_stable: SqStable(P)
, 
implies: P 
⇒ Q
, 
squash: ↓T
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
req_int_terms: t1 ≡ t2
, 
false: False
, 
not: ¬A
, 
top: Top
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
rmul_wf, 
rleq_wf, 
radd_wf, 
req_wf, 
int-to-real_wf, 
metric_wf, 
real_wf, 
istype-universe, 
rmul_preserves_rleq2, 
sq_stable__rleq, 
itermSubtract_wf, 
itermMultiply_wf, 
itermAdd_wf, 
itermVar_wf, 
rleq-implies-rleq, 
rsub_wf, 
req-iff-rsub-is-0, 
rleq_functionality, 
req_weakening, 
real_polynomial_null, 
istype-int, 
real_term_value_sub_lemma, 
istype-void, 
real_term_value_mul_lemma, 
real_term_value_add_lemma, 
real_term_value_var_lemma, 
real_term_value_const_lemma, 
rmul-zero, 
req_functionality, 
rmul_functionality
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
dependent_set_memberEquality_alt, 
lambdaEquality_alt, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
inhabitedIsType, 
universeIsType, 
sqequalRule, 
productElimination, 
lambdaFormation_alt, 
independent_pairFormation, 
because_Cache, 
productIsType, 
functionIsType, 
natural_numberEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
setIsType, 
instantiate, 
universeEquality, 
independent_isectElimination, 
independent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
dependent_functionElimination, 
approximateComputation, 
int_eqEquality, 
voidElimination
Latex:
\mforall{}[X:Type].  \mforall{}[c:\{c:\mBbbR{}|  r0  \mleq{}  c\}  ].  \mforall{}[d:metric(X)].    (c*d  \mmember{}  metric(X))
Date html generated:
2019_10_29-AM-11_06_12
Last ObjectModification:
2019_10_02-AM-09_47_47
Theory : reals
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