Nuprl Lemma : test-eq-product
∀[A,B1,B2,C,D:Type]. ∀[a1,a2:A]. ∀[b1:B1]. ∀[b1':B2]. ∀[b2:B1]. ∀[b2':B2]. ∀[c1,c2:C]. ∀[d1,d2:D].
(<a1, <b1, b1'>, c1, d1> = <a2, <b2, b2'>, c2, d2> ∈ (a:A × b:B1 × B2 × c:C × D)
⇐⇒ (a1 = a2 ∈ A) ∧ ((b1 = b2 ∈ B1) ∧ (b1' = b2' ∈ B2)) ∧ (c1 = c2 ∈ C) ∧ (d1 = d2 ∈ D))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
pi1: fst(t),
member: t ∈ T,
pi2: snd(t),
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
uall_wf,
iff_wf
Rules used in proof :
cut,
addLevel,
uallFunctionality,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
applyLambdaEquality,
sqequalRule,
hypothesisEquality,
hypothesis,
cumulativity,
introduction,
extract_by_obid,
isectElimination,
productEquality,
dependent_pairEquality,
independent_pairEquality,
instantiate,
universeEquality,
lambdaEquality,
because_Cache,
applyEquality,
isect_memberFormation,
lambdaFormation,
dependent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A,B1,B2,C,D:Type]. \mforall{}[a1,a2:A]. \mforall{}[b1:B1]. \mforall{}[b1':B2]. \mforall{}[b2:B1]. \mforall{}[b2':B2]. \mforall{}[c1,c2:C]. \mforall{}[d1,d2:D].
(<a1, <b1, b1'>, c1, d1> = <a2, <b2, b2'>, c2, d2>
\mLeftarrow{}{}\mRightarrow{} (a1 = a2) \mwedge{} ((b1 = b2) \mwedge{} (b1' = b2')) \mwedge{} (c1 = c2) \mwedge{} (d1 = d2))
Date html generated:
2017_10_01-AM-09_11_14
Last ObjectModification:
2017_07_26-PM-04_47_20
Theory : general
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