Nuprl Lemma : compose-mfun
∀[X,Y,Z:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[dZ:metric(Z)]. ∀[f:FUN(X ⟶ Y)]. ∀[g:FUN(Y ⟶ Z)].
(g o f ∈ FUN(X ⟶ Z))
Proof
Definitions occuring in Statement :
mfun: FUN(X ⟶ Y),
metric: metric(X),
compose: f o g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
mfun: FUN(X ⟶ Y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
is-mfun: f:FUN(X;Y),
so_apply: x[s],
compose: f o g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
compose_wf,
meq_wf,
is-mfun_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setIsType,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
applyEquality
Latex:
\mforall{}[X,Y,Z:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. \mforall{}[dZ:metric(Z)]. \mforall{}[f:FUN(X {}\mrightarrow{} Y)].
\mforall{}[g:FUN(Y {}\mrightarrow{} Z)].
(g o f \mmember{} FUN(X {}\mrightarrow{} Z))
Date html generated:
2019_10_30-AM-06_22_13
Last ObjectModification:
2019_10_02-AM-09_58_01
Theory : reals
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