Nuprl Lemma : free-vs-dim-0
∀S:Type. ((¬S) ⇒ (∀K:CRng. free-vs(K;S) ≅ 0))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-iso: A ≅ B,
trivial-vs: 0,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type,
crng: CRng
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
iff: P ⇐⇒ Q,
and: P ∧ Q,
vs-point: Point(vs),
record-select: r.x,
trivial-vs: 0,
mk-vs: mk-vs,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
unit: Unit,
exists: ∃x:A. B[x],
not: ¬A,
false: False,
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
exists!: ∃!x:T. P[x],
vs-map: A ⟶ B,
cand: A c∧ B,
vs-add: x + y,
uimplies: b supposing a,
vs-0: 0
Lemmas referenced :
free-vs-unique,
trivial-vs_wf,
istype-void,
vector-space_wf,
exists!_wf,
vs-map_wf,
all_wf,
equal_wf,
vs-point_wf,
vs-iso_inversion,
free-vs_wf,
crng_wf,
istype-universe,
vs-0_wf,
rec_select_update_lemma,
unit_wf2,
vs-zero-add,
vs-zero-mul,
rng_car_wf,
it_wf,
vs-add_wf,
vs-mul_wf,
vs-map-eq,
vs-map-0,
equal-unit
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
dependent_pairFormation_alt,
functionExtensionality,
applyEquality,
voidElimination,
isect_memberEquality_alt,
functionIsType,
because_Cache,
universeIsType,
lambdaEquality_alt,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
equalitySymmetry,
inhabitedIsType,
independent_pairFormation,
productIsType,
equalityIstype,
independent_isectElimination,
equalityTransitivity
Latex:
\mforall{}S:Type. ((\mneg{}S) {}\mRightarrow{} (\mforall{}K:CRng. free-vs(K;S) \mcong{} 0))
Date html generated:
2019_10_31-AM-06_30_44
Last ObjectModification:
2019_08_02-PM-03_52_02
Theory : linear!algebra
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