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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