Nuprl Lemma : fs-in-subtype-subspace
∀[K:CRng]. ∀[S,T:Type]. vs-subspace(K;free-vs(K;S);f.fs-in-subtype(K;S;T;f)) supposing strong-subtype(T;S)
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
fs-in-subtype: fs-in-subtype(K;S;T;f),
vs-subspace: vs-subspace(K;vs;x.P[x]),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
vs-subspace: vs-subspace(K;vs;x.P[x]),
fs-in-subtype: fs-in-subtype(K;S;T;f),
fs-predicate: fs-predicate(K;S;p.P[p];f),
squash: ↓T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
cand: A c∧ B,
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
vs-point: Point(vs),
record-select: r.x,
free-vs: free-vs(K;S),
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,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
vs-0: 0,
empty-bag: {},
nil: [],
it: ⋅,
basic-formal-sum: basic-formal-sum(K;S),
exists: ∃x:A. B[x],
bfs-predicate: bfs-predicate(K;S;p.P[p];b),
pi2: snd(t),
false: False,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
respects-equality: respects-equality(S;T),
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
vs-add: x + y,
formal-sum-add: x + y,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
sq_or: a ↓∨ b,
or: P ∨ Q,
formal-sum-mul: k * x,
bag-map: bag-map(f;bs),
map: map(f;as),
vs-mul: a * x,
infix_ap: x f y,
top: Top,
pi1: fst(t)
Lemmas referenced :
strong-subtype-iff-respects-equality,
strong-subtype_wf,
istype-universe,
crng_wf,
fs-in-subtype_wf,
subtype_rel_self,
formal-sum_wf,
vs-point_wf,
free-vs_wf,
rng_car_wf,
equal-wf,
empty-bag_wf,
bag-member-empty-iff,
bag-member_wf,
respects-equality-quotient1,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
respects-equality-trivial,
bfs-predicate_wf,
pi2_wf,
bag-append_wf,
vs-add_wf,
subtype_quotient,
squash_wf,
true_wf,
vector-space_wf,
rng_sig_wf,
equal_functionality_wrt_subtype_rel2,
bag-member-append,
trivial-equal,
member_wf,
formal-sum-mul_wf1,
vs-mul_wf,
bag_wf,
bag-member-map,
rng_times_wf,
pi1_wf_top,
istype-void,
strong-subtype-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
independent_pairFormation,
lambdaFormation_alt,
setElimination,
rename,
applyEquality,
because_Cache,
independent_functionElimination,
productEquality,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
voidElimination,
productIsType,
equalityIstype,
sqequalBase,
natural_numberEquality,
unionElimination,
hyp_replacement,
spreadEquality,
applyLambdaEquality
Latex:
\mforall{}[K:CRng]. \mforall{}[S,T:Type].
vs-subspace(K;free-vs(K;S);f.fs-in-subtype(K;S;T;f)) supposing strong-subtype(T;S)
Date html generated:
2019_10_31-AM-06_30_04
Last ObjectModification:
2019_08_19-PM-01_09_06
Theory : linear!algebra
Home
Index