Nuprl Lemma : vs-map-compose
∀[K:RngSig]. ∀[A,B,C:VectorSpace(K)]. ∀[f:A ⟶ B]. ∀[g:B ⟶ C]. (g o f ∈ A ⟶ C)
Proof
Definitions occuring in Statement :
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
compose: f o g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng_sig: RngSig
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
cand: A c∧ B,
compose: f o g,
and: P ∧ Q,
vs-map: A ⟶ B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_sig_wf,
vector-space_wf,
vs-map_wf,
all_wf,
rng_car_wf,
vs-mul_wf,
iff_weakening_equal,
vs-add_wf,
true_wf,
squash_wf,
equal_wf,
vs-point_wf,
compose_wf
Rules used in proof :
isect_memberEquality,
axiomEquality,
productEquality,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
lambdaFormation,
sqequalRule,
applyEquality,
functionExtensionality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
productElimination,
dependent_set_memberEquality,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:RngSig]. \mforall{}[A,B,C:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} C]. (g o f \mmember{} A {}\mrightarrow{} C)
Date html generated:
2018_05_22-PM-09_42_47
Last ObjectModification:
2018_01_09-AM-10_49_45
Theory : linear!algebra
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