Nuprl Lemma : equal-iff-vs-subtract-is-0
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[x,y:Point(vs)]. (x = y ∈ Point(vs) ⇐⇒ (x - y) = 0 ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-subtract: (x - y),
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
rng: Rng,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
vs-subtract: (x - y),
vs-neg: -(x)
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-point_wf,
vs-subtract-self,
iff_weakening_equal,
vs-0_wf,
vs-subtract_wf,
equal_wf,
vs-zero-add,
rng_sig_wf,
true_wf,
squash_wf,
vs-add_wf,
vs-grp_inv_assoc,
vs-add-comm,
vs-ac_1,
rng_one_wf,
rng_minus_wf,
vs-mul_wf,
vs-mon_assoc
Rules used in proof :
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
independent_pairEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
equalitySymmetry,
hypothesisEquality,
rename,
setElimination,
levelHypothesis,
equalityUniverse,
hypothesis,
because_Cache,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
universeEquality,
hyp_replacement
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[x,y:Point(vs)]. (x = y \mLeftarrow{}{}\mRightarrow{} (x - y) = 0)
Date html generated:
2018_05_22-PM-09_42_59
Last ObjectModification:
2018_01_09-PM-02_27_58
Theory : linear!algebra
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