Nuprl Lemma : vs-map-quotients
∀[K:CRng]. ∀[A,B:VectorSpace(K)]. ∀[P:Point(A) ⟶ ℙ]. ∀[Q:Point(B) ⟶ ℙ].
∀[f:A ⟶ B]. f ∈ A//z.P[z] ⟶ B//z.Q[z] supposing ∀a:Point(A). (P[a] ⇒ Q[f a])
supposing vs-subspace(K;A;z.P[z]) ∧ vs-subspace(K;B;z.Q[z])
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
vs-map: A ⟶ B,
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],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
vs-map: A ⟶ B,
and: P ∧ Q,
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
eq-mod-subspace: x = y mod (z.P[z]),
quotient: x,y:A//B[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
mk-vs: mk-vs,
vs-point: Point(vs),
vs-quotient: vs//z.P[z],
guard: {T},
true: True,
vs-neg: -(x),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
vs-add: x + y,
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-mul: a * x
Lemmas referenced :
vs-point_wf,
vs-quotient_wf,
vs-add_wf,
rng_car_wf,
vs-mul_wf,
subtype_rel_self,
vs-map_wf,
vs-subspace_wf,
vector-space_wf,
crng_wf,
quotient-member-eq,
eq-mod-subspace-equiv,
eq-mod-subspace_wf,
quotient_wf,
istype-void,
rec_select_update_lemma,
vs-neg_wf,
rng_one_wf,
rng_minus_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rng_sig_wf,
iff_weakening_equal,
vs-add-assoc,
vs-neg-add2,
vs-ac_1,
vs-add-comm-nu,
vs-mul-linear,
crng_times_comm,
vs-mul-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
equalityTransitivity,
hypothesis,
equalitySymmetry,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
productElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
independent_isectElimination,
equalityIstype,
instantiate,
universeEquality,
dependent_functionElimination,
functionExtensionality,
sqequalBase,
promote_hyp,
pertypeElimination,
independent_functionElimination,
inhabitedIsType,
pointwiseFunctionalityForEquality,
voidElimination,
isect_memberEquality_alt,
applyLambdaEquality,
hyp_replacement,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaFormation_alt,
independent_pairFormation,
Error :memTop
Latex:
\mforall{}[K:CRng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[P:Point(A) {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:Point(B) {}\mrightarrow{} \mBbbP{}].
\mforall{}[f:A {}\mrightarrow{} B]. f \mmember{} A//z.P[z] {}\mrightarrow{} B//z.Q[z] supposing \mforall{}a:Point(A). (P[a] {}\mRightarrow{} Q[f a])
supposing vs-subspace(K;A;z.P[z]) \mwedge{} vs-subspace(K;B;z.Q[z])
Date html generated:
2020_05_20-PM-01_18_26
Last ObjectModification:
2020_01_03-AM-00_51_15
Theory : linear!algebra
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