Nuprl Lemma : compose-injections
∀[T:Type]. ∀[f,g:{f:T ⟶ T| Inj(T;T;f)} ]. (f o g ∈ {f:T ⟶ T| Inj(T;T;f)} )
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
compose: f o g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
compose: f o g,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
compose_wf,
equal_wf,
inject_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
lambdaFormation,
sqequalRule,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaEquality,
isect_memberEquality,
because_Cache,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[f,g:\{f:T {}\mrightarrow{} T| Inj(T;T;f)\} ]. (f o g \mmember{} \{f:T {}\mrightarrow{} T| Inj(T;T;f)\} )
Date html generated:
2016_05_13-PM-04_06_08
Last ObjectModification:
2015_12_26-AM-11_04_49
Theory : fun_1
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