Nuprl Lemma : pair-mono
∀A:Type. ∀B:A ⟶ Type. ((mono(A) ∧ (∀a:A. mono(B[a]))) ⇒ mono(a:A × B[a]))
Proof
Definitions occuring in Statement :
mono: mono(T),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: 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,
and: P ∧ Q,
mono: mono(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
guard: {T}
Lemmas referenced :
is-above-pair,
is-above_wf,
base_wf,
and_wf,
mono_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
dependent_pairEquality,
productEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. ((mono(A) \mwedge{} (\mforall{}a:A. mono(B[a]))) {}\mRightarrow{} mono(a:A \mtimes{} B[a]))
Date html generated:
2016_05_13-PM-04_13_42
Last ObjectModification:
2015_12_26-AM-11_10_26
Theory : subtype_1
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