Nuprl Lemma : member-per-product
∀[A:Type]. ∀[B:per-function(A;a.Type)]. ∀[a:A]. ∀[b:B[a]]. (<a, b> ∈ per-product(A;a.B[a]))
Proof
Definitions occuring in Statement :
per-product: per-product(A;a.B[a]),
per-function: per-function(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
pair: <a, b>,
universe: Type
Definitions unfolded in proof :
type-function: type-function{i:l}(A),
so_apply: x[s],
member: t ∈ T,
uall: ∀[x:A]. B[x],
per-product: per-product(A;a.B[a]),
squash: ↓T,
prop: ℙ,
pi1: fst(t),
pi2: snd(t),
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uand: uand(A;B),
has-value: (a)↓,
top: Top,
and: P ∧ Q
Lemmas referenced :
apply_wf_type-function,
per-product_wf,
per-function_wf_type,
member_wf,
squash_wf,
true_wf,
equal_wf,
subtype_rel_self,
equal-wf-base,
has-value_wf_base,
is-exception_wf,
top_wf,
base_wf,
and_wf
Rules used in proof :
universeEquality,
sqequalRule,
equalitySymmetry,
hypothesis,
equalityTransitivity,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
pointwiseFunctionality,
applyEquality,
lambdaEquality,
imageElimination,
pertypeMemberEquality,
baseApply,
closedConclusion,
baseClosed,
natural_numberEquality,
imageMemberEquality,
instantiate,
axiomSqEquality,
axiomEquality,
isect_memberEquality,
axiomSqleEquality,
rename,
isaxiomCases,
divergentSqle,
voidElimination,
voidEquality,
hyp_replacement,
productElimination,
setElimination,
applyLambdaEquality,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:per-function(A;a.Type)]. \mforall{}[a:A]. \mforall{}[b:B[a]]. (<a, b> \mmember{} per-product(A;a.B[a]))
Date html generated:
2019_06_20-AM-11_30_20
Last ObjectModification:
2018_08_04-PM-03_40_18
Theory : per!type
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