Nuprl Lemma : induced-metric_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[Y:Type]. ∀[f:Y ⟶ X]. (induced-metric(d;f) ∈ metric(Y))
Proof
Definitions occuring in Statement :
induced-metric: induced-metric(d;f),
metric: metric(X),
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
metric: metric(X),
induced-metric: induced-metric(d;f),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
prop: ℙ,
guard: {T}
Lemmas referenced :
rleq_wf,
int-to-real_wf,
req_wf,
radd_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
inhabitedIsType,
universeIsType,
dependent_set_memberEquality_alt,
equalityTransitivity,
hypothesis,
equalitySymmetry,
productElimination,
lambdaFormation_alt,
independent_pairFormation,
because_Cache,
productIsType,
functionIsType,
extract_by_obid,
isectElimination,
natural_numberEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[Y:Type]. \mforall{}[f:Y {}\mrightarrow{} X]. (induced-metric(d;f) \mmember{} metric(Y))
Date html generated:
2019_10_29-AM-11_04_11
Last ObjectModification:
2019_10_02-AM-09_46_03
Theory : reals
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