Nuprl Lemma : free-vs-map-into-subspace
∀[K:CRng]. ∀[S:Type]. ∀[v:VectorSpace(K)]. ∀[f:free-vs(K;S) ⟶ v]. ∀[P:Point(v) ⟶ ℙ].
(f ∈ free-vs(K;S) ⟶ (x:v | P[x])) supposing ((∀s:S. (↓P[f <s>])) and vs-subspace(K;v;x.P[x]))
Proof
Definitions occuring in Statement :
free-vs-inc: <s>,
free-vs: free-vs(K;S),
vs-map: A ⟶ B,
sub-vs: (v:vs | P[v]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
so_apply: x[s],
vs-map: A ⟶ B,
crng: CRng,
rng: Rng,
prop: ℙ,
so_lambda: λ2x.t[x],
free-vs: free-vs(K;S),
vs-point: Point(vs),
mk-vs: mk-vs,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
sub-vs: (v:vs | P[v]),
squash: ↓T,
subtype_rel: A ⊆r B,
record-select: r.x,
record-update: r[x := v],
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
basic-formal-sum: basic-formal-sum(K;S),
vs-lift: vs-lift(vs;f;fs),
exists: ∃x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
decidable: Dec(P),
vs-bag-add: Σ{f[b] | b ∈ bs},
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
vs-0: 0,
vs-subspace: vs-subspace(K;vs;x.P[x]),
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
single-bag: {x},
vs-mul: a * x,
vs-add: x + y
Lemmas referenced :
squash_wf,
free-vs-inc_wf,
vs-subspace_wf,
vs-point_wf,
vs-map_wf,
free-vs_wf,
vector-space_wf,
istype-universe,
crng_wf,
vs-lift-unique,
rec_select_update_lemma,
istype-void,
sub-vs_wf,
basic-formal-sum_wf,
bfs-equiv_wf,
subtype_rel_self,
subtype_quotient,
bfs-equiv-rel,
iff_weakening_equal,
bag_to_squash_list,
rng_car_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
vs-bag-add_wf,
vs-mul_wf,
squash-test,
vs-0_wf,
vs-bag-add-append,
single-bag_wf,
list-subtype-bag,
list_accum_cons_lemma,
list_accum_nil_lemma,
vs-mon_ident,
vs-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
functionIsType,
universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
thin,
applyEquality,
setElimination,
rename,
because_Cache,
lambdaEquality_alt,
universeEquality,
dependent_functionElimination,
instantiate,
independent_isectElimination,
lambdaFormation_alt,
isect_memberEquality_alt,
voidElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
productElimination,
productIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
imageElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
inhabitedIsType,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
productEquality,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
baseApply,
closedConclusion,
intEquality,
hyp_replacement,
spreadEquality,
functionExtensionality
Latex:
\mforall{}[K:CRng]. \mforall{}[S:Type]. \mforall{}[v:VectorSpace(K)]. \mforall{}[f:free-vs(K;S) {}\mrightarrow{} v]. \mforall{}[P:Point(v) {}\mrightarrow{} \mBbbP{}].
(f \mmember{} free-vs(K;S) {}\mrightarrow{} (x:v | P[x])) supposing ((\mforall{}s:S. (\mdownarrow{}P[f <s>])) and vs-subspace(K;v;x.P[x]))
Date html generated:
2019_10_31-AM-06_31_31
Last ObjectModification:
2019_08_19-PM-05_21_26
Theory : linear!algebra
Home
Index