Nuprl Lemma : prod2-metric-meq
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[p,q:X × Y]. uiff(p ≡ q;fst(p) ≡ fst(q) ∧ snd(p) ≡ snd(q))
Proof
Definitions occuring in Statement :
prod2-metric: prod2-metric(dX;dY),
meq: x ≡ y,
metric: metric(X),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
meq: x ≡ y,
metric: metric(X),
prod2-metric: prod2-metric(dX;dY),
pi1: fst(t),
pi2: snd(t),
mdist: mdist(d;x;y),
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T
Lemmas referenced :
sq_stable__uiff,
meq_wf,
prod2-metric_wf,
pi1_wf_top,
istype-void,
pi2_wf,
sq_stable__meq,
sq_stable__and,
req_witness,
int-to-real_wf,
mdist_wf,
req_wf,
iff_weakening_uiff,
radd_wf,
radd-of-nonneg-is-zero,
mdist-nonneg,
rleq_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
lambdaEquality_alt,
universeIsType,
independent_functionElimination,
lambdaFormation_alt,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
natural_numberEquality,
functionIsTypeImplies,
inhabitedIsType,
independent_pairFormation,
because_Cache,
productIsType,
independent_isectElimination,
dependent_set_memberEquality_alt,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
universeEquality
Latex:
\mforall{}[X,Y:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. \mforall{}[p,q:X \mtimes{} Y].
uiff(p \mequiv{} q;fst(p) \mequiv{} fst(q) \mwedge{} snd(p) \mequiv{} snd(q))
Date html generated:
2019_10_29-AM-11_10_55
Last ObjectModification:
2019_10_02-AM-09_51_36
Theory : reals
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