Nuprl Lemma : subtype_rel_dep_product_iff
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:Type]. ∀[D:C ⟶ Type].
uiff(∀[a:A]. (B[a] ⊆r D[a]);(a:A × B[a]) ⊆r (c:C × D[c])) supposing A ⊆r C
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
subtype_rel_product,
uall_wf,
subtype_rel_wf,
pi2_wf,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
hypothesis,
lambdaFormation,
because_Cache,
axiomEquality,
isect_memberEquality,
productEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
dependent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:Type]. \mforall{}[D:C {}\mrightarrow{} Type].
uiff(\mforall{}[a:A]. (B[a] \msubseteq{}r D[a]);(a:A \mtimes{} B[a]) \msubseteq{}r (c:C \mtimes{} D[c])) supposing A \msubseteq{}r C
Date html generated:
2016_05_13-PM-03_18_43
Last ObjectModification:
2015_12_26-AM-09_08_27
Theory : subtype_0
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