Nuprl Lemma : vs-add_functionality_eq-mod
∀K:Rng. ∀vs:VectorSpace(K). ∀P:Point(vs) ⟶ ℙ.
(vs-subspace(K;vs;z.P[z])
⇒ (∀x,y,x',y':Point(vs). (x = x' mod (z.P[z]) ⇒ y = y' mod (z.P[z]) ⇒ x + y = x' + y' mod (z.P[z]))))
Proof
Definitions occuring in Statement :
eq-mod-subspace: x = y mod (z.P[z]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
eq-mod-subspace: x = y mod (z.P[z]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
so_lambda: λ2x.t[x]
Lemmas referenced :
vs-add_wf,
vs-neg_wf,
vs-add-assoc,
subtype_rel_self,
iff_weakening_equal,
squash_wf,
true_wf,
vs-point_wf,
vector-space_wf,
rng_sig_wf,
vs-neg-add2,
vs-ac_1,
vs-add-comm-nu,
eq-mod-subspace_wf,
vs-subspace_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesisEquality,
independent_functionElimination,
applyEquality,
sqequalRule,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionIsType
Latex:
\mforall{}K:Rng. \mforall{}vs:VectorSpace(K). \mforall{}P:Point(vs) {}\mrightarrow{} \mBbbP{}.
(vs-subspace(K;vs;z.P[z])
{}\mRightarrow{} (\mforall{}x,y,x',y':Point(vs).
(x = x' mod (z.P[z]) {}\mRightarrow{} y = y' mod (z.P[z]) {}\mRightarrow{} x + y = x' + y' mod (z.P[z]))))
Date html generated:
2020_05_20-PM-01_18_09
Last ObjectModification:
2020_01_03-AM-00_51_00
Theory : linear!algebra
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