Nuprl Lemma : equal-product1
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[a1,a2:A]. ∀[b1:B[a1]]. ∀[b2:B[a2]].
{<a1, b1> = <a2, b2> ∈ (a:A × B[a]) ⇐⇒ (a1 = a2 ∈ A) ∧ (b1 = b2 ∈ B[a1])}
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
pi1: fst(t),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
pi2: snd(t),
uimplies: b supposing a
Lemmas referenced :
and_wf,
equal_wf,
pi1_wf,
subtype_rel_self,
subtype_rel_wf,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesisEquality,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
equalityTransitivity,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
applyEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality,
cumulativity,
functionExtensionality,
dependent_pairEquality,
hyp_replacement,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[a1,a2:A]. \mforall{}[b1:B[a1]]. \mforall{}[b2:B[a2]].
\{<a1, b1> = <a2, b2> \mLeftarrow{}{}\mRightarrow{} (a1 = a2) \mwedge{} (b1 = b2)\}
Date html generated:
2017_10_01-AM-09_11_06
Last ObjectModification:
2017_07_26-PM-04_47_16
Theory : general
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