Nuprl Lemma : prod2-metric_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (prod2-metric(dX;dY) ∈ metric(X × Y))
Proof
Definitions occuring in Statement :
prod2-metric: prod2-metric(dX;dY),
metric: metric(X),
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
metric: metric(X),
prod2-metric: prod2-metric(dX;dY),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
radd_wf,
mdist_wf,
radd-non-neg,
mdist-nonneg,
rleq_wf,
int-to-real_wf,
req_wf,
metric_wf,
istype-universe,
radd-zero,
req_functionality,
radd_functionality,
mdist-same,
req_weakening,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
radd_functionality_wrt_rleq,
mdist-triangle-inequality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality,
mdist-symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
spreadEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
inhabitedIsType,
productIsType,
universeIsType,
lambdaFormation_alt,
sqequalRule,
productElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
functionIsType,
natural_numberEquality,
applyEquality,
instantiate,
universeEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[X,Y:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. (prod2-metric(dX;dY) \mmember{} metric(X \mtimes{} Y))
Date html generated:
2019_10_29-AM-11_10_36
Last ObjectModification:
2019_10_02-AM-09_51_18
Theory : reals
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