Nuprl Lemma : subtype_rel_product
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:Type]. ∀[D:C ⟶ Type].
((a:A × B[a]) ⊆r (c:C × D[c])) supposing ((∀a:A. (B[a] ⊆r D[a])) and (A ⊆r C))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
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,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
all_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
productElimination,
thin,
dependent_pairEquality,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
dependent_functionElimination,
productEquality,
axiomEquality,
lemma_by_obid,
isectElimination,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:Type]. \mforall{}[D:C {}\mrightarrow{} Type].
((a:A \mtimes{} B[a]) \msubseteq{}r (c:C \mtimes{} D[c])) supposing ((\mforall{}a:A. (B[a] \msubseteq{}r D[a])) and (A \msubseteq{}r C))
Date html generated:
2016_05_13-PM-03_18_39
Last ObjectModification:
2015_12_26-AM-09_08_25
Theory : subtype_0
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