Nuprl Lemma : prod-metric-meq
∀[k:ℕ]. ∀[X:ℕk ⟶ Type]. ∀[d:i:ℕk ⟶ metric(X[i])]. ∀[p,q:i:ℕk ⟶ X[i]]. uiff(p ≡ q;∀i:ℕk. p i ≡ q i)
Proof
Definitions occuring in Statement :
prod-metric: prod-metric(k;d),
meq: x ≡ y,
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
meq: x ≡ y,
subtype_rel: A ⊆r B,
metric: metric(X),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
prod-metric: prod-metric(k;d),
mdist: mdist(d;x;y),
squash: ↓T,
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
sq_stable__uiff,
meq_wf,
int_seg_wf,
prod-metric_wf,
sq_stable__meq,
sq_stable__all,
req_witness,
int-to-real_wf,
metric_wf,
istype-universe,
istype-nat,
rsum-of-nonneg-zero-iff,
subtract_wf,
subtract-add-cancel,
mdist_wf,
mdist-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
universeIsType,
independent_functionElimination,
lambdaFormation_alt,
dependent_functionElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
functionIsType,
instantiate,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} Type]. \mforall{}[d:i:\mBbbN{}k {}\mrightarrow{} metric(X[i])]. \mforall{}[p,q:i:\mBbbN{}k {}\mrightarrow{} X[i]].
uiff(p \mequiv{} q;\mforall{}i:\mBbbN{}k. p i \mequiv{} q i)
Date html generated:
2019_10_29-AM-11_09_54
Last ObjectModification:
2019_10_10-PM-10_07_38
Theory : reals
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