Nuprl Lemma : exists-product3
∀[A,B,C,D:Type]. ∀P:(A × B × C × D) ⟶ ℙ'. {∃x:A × B × C × D. P[x] ⇐⇒ ∃a:A. ∃b:B. ∃c:C. ∃d:D. P[<a, b, c, d>]}
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
exists_wf,
exists-product1,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
independent_pairEquality,
because_Cache,
addLevel,
productElimination,
independent_functionElimination,
productEquality,
dependent_functionElimination,
existsFunctionality,
impliesFunctionality,
levelHypothesis,
existsLevelFunctionality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B,C,D:Type].
\mforall{}P:(A \mtimes{} B \mtimes{} C \mtimes{} D) {}\mrightarrow{} \mBbbP{}'. \{\mexists{}x:A \mtimes{} B \mtimes{} C \mtimes{} D. P[x] \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. \mexists{}b:B. \mexists{}c:C. \mexists{}d:D. P[<a, b, c, d>]\}
Date html generated:
2016_05_15-PM-03_23_02
Last ObjectModification:
2015_12_27-PM-01_05_41
Theory : general
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