Nuprl Lemma : finite-product
∀[S:Type]. ∀[T:S ⟶ Type]. (finite(S) ⇒ (∀s:S. finite(T[s])) ⇒ finite(s:S × T[s]))
Proof
Definitions occuring in Statement :
finite: finite(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
finite: finite(T),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
pi1: fst(t),
equipollent: A ~ B,
inv_funs: InvFuns(A;B;f;g),
and: P ∧ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
tidentity: Id{T},
compose: f o g,
identity: Id,
true: True,
guard: {T},
squash: ↓T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
finite_wf,
istype-universe,
equipollent_wf,
int_seg_wf,
istype-nat,
bij_imp_exists_inv,
product_functionality_wrt_equipollent_dependent,
squash_wf,
true_wf,
istype-int,
iff_weakening_equal,
equipollent_functionality_wrt_equipollent2,
sum_wf,
equipollent-sum,
sum-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
hypothesis,
promote_hyp,
functionIsType,
universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
inhabitedIsType,
instantiate,
universeEquality,
because_Cache,
functionExtensionality,
natural_numberEquality,
lambdaEquality_alt,
setElimination,
rename,
dependent_pairFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productEquality
Latex:
\mforall{}[S:Type]. \mforall{}[T:S {}\mrightarrow{} Type]. (finite(S) {}\mRightarrow{} (\mforall{}s:S. finite(T[s])) {}\mRightarrow{} finite(s:S \mtimes{} T[s]))
Date html generated:
2020_05_19-PM-10_00_42
Last ObjectModification:
2019_10_25-PM-09_02_51
Theory : equipollence!!cardinality!
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