Nuprl Lemma : product-discrete
∀A:Type. ∀B:A ⟶ Type. (discrete-type(A) ⇒ (∀a:A. discrete-type(B[a])) ⇒ discrete-type(a:A × B[a]))
Proof
Definitions occuring in Statement :
discrete-type: discrete-type(T),
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
discrete-type: discrete-type(T),
member: t ∈ T,
so_apply: x[s],
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
real_wf,
pi1_wf_top,
equal_wf,
req_wf,
all_wf,
discrete-type_wf,
squash_wf,
true_wf,
pair_eta_rw,
iff_weakening_equal,
subtype_rel-equal,
and_wf,
subtype_rel_product,
top_wf,
pi2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
productEquality,
cumulativity,
isectElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_functionElimination,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality,
universeEquality,
dependent_pairEquality,
independent_isectElimination,
hyp_replacement,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type.
(discrete-type(A) {}\mRightarrow{} (\mforall{}a:A. discrete-type(B[a])) {}\mRightarrow{} discrete-type(a:A \mtimes{} B[a]))
Date html generated:
2018_05_22-PM-02_14_09
Last ObjectModification:
2017_10_29-PM-08_05_01
Theory : reals
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