Nuprl Lemma : is-short-exact_wf
∀[K:Rng]. ∀[A,B,C:VectorSpace(K)]. ∀[f:A ⟶ B]. ∀[g:B ⟶ C]. (is-short-exact(A;B;C;f;g) ∈ ℙ)
Proof
Definitions occuring in Statement :
is-short-exact: is-short-exact(A;B;C;f;g),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
is-short-exact: is-short-exact(A;B;C;f;g),
prop: ℙ,
and: P ∧ Q,
all: ∀x:A. B[x],
rng: Rng,
vs-map: A ⟶ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
vs-point_wf,
iff_wf,
vs-map-kernel_wf,
equal_wf,
vs-0_wf,
vs-map-image_wf,
vs-map_wf,
vector-space_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
productEquality,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B,C:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} C]. (is-short-exact(A;B;C;f;g) \mmember{} \mBbbP{})
Date html generated:
2019_10_31-AM-06_27_40
Last ObjectModification:
2019_08_21-PM-06_33_04
Theory : linear!algebra
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