Nuprl Lemma : null-formal-sum-neg
∀K:Rng. ∀[S:Type]. ∀[fs:basic-formal-sum(K;S)]. (null-formal-sum(K;S;fs) ⇒ null-formal-sum(K;S;-(fs)))
Proof
Definitions occuring in Statement :
null-formal-sum: null-formal-sum(K;S;fs),
neg-bfs: -(fs),
basic-formal-sum: basic-formal-sum(K;S),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
rng: Rng
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
squash: ↓T,
subtype_rel: A ⊆r B,
basic-formal-sum: basic-formal-sum(K;S),
true: True,
rng: Rng,
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
null-formal-sum: null-formal-sum(K;S;fs),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
compose: f o g,
top: Top,
neg-bfs: -(fs),
zero-bfs: 0 * ss
Lemmas referenced :
exists_wf,
equal_wf,
zero-bfs_wf,
bag_wf,
neg-bfs_wf,
rng_car_wf,
bag-append_wf,
rng_wf,
basic-formal-sum_wf,
null-formal-sum_wf,
squash_wf,
true_wf,
rng_sig_wf,
neg-bfs-append,
iff_weakening_equal,
bag-map_wf,
bag-map-map,
rng_zero_wf,
rng_minus_zero
Rules used in proof :
baseClosed,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
lambdaEquality,
applyEquality,
sqequalRule,
productEquality,
natural_numberEquality,
because_Cache,
universeEquality,
cumulativity,
hypothesis,
hypothesisEquality,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
functionEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairEquality,
functionExtensionality
Latex:
\mforall{}K:Rng
\mforall{}[S:Type]. \mforall{}[fs:basic-formal-sum(K;S)]. (null-formal-sum(K;S;fs) {}\mRightarrow{} null-formal-sum(K;S;-(fs)))
Date html generated:
2018_05_22-PM-09_47_16
Last ObjectModification:
2018_01_08-PM-01_29_00
Theory : linear!algebra
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