Nuprl Lemma : product_subtype_base
∀[A:Type]. ∀[B:A ⟶ Type].  ((a:A × B[a]) ⊆r Base) supposing ((∀a:A. (B[a] ⊆r Base)) and (A ⊆r Base))
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]
, 
base: Base
, 
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 : 
base_wf, 
subtype_rel_wf, 
all_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaEquality, 
productElimination, 
thin, 
sqequalRule, 
baseApply, 
closedConclusion, 
baseClosed, 
hypothesisEquality, 
applyEquality, 
hypothesis, 
sqequalHypSubstitution, 
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].
    ((a:A  \mtimes{}  B[a])  \msubseteq{}r  Base)  supposing  ((\mforall{}a:A.  (B[a]  \msubseteq{}r  Base))  and  (A  \msubseteq{}r  Base))
Date html generated:
2016_05_13-PM-03_19_23
Last ObjectModification:
2016_01_14-PM-04_32_02
Theory : subtype_0
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