Nuprl Lemma : formal-sum-mul_functionality
∀S:Type. ∀K:CRng. ∀x,x':basic-formal-sum(K;S). ∀k,k':|K|.
bfs-equiv(K;S;x;x') ⇒ bfs-equiv(K;S;k * x;k' * x') supposing k = k' ∈ |K|
Proof
Definitions occuring in Statement :
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
formal-sum-mul: k * x,
basic-formal-sum: basic-formal-sum(K;S),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
true: True,
so_lambda: λ2x y.t[x; y],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bfs-reduce: bfs-reduce(K;S;as;bs),
or: P ∨ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
basic-formal-sum: basic-formal-sum(K;S),
infix_ap: x f y,
so_apply: x[s],
formal-sum-mul: k * x,
top: Top,
zero-bfs: 0 * ss,
compose: f o g,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
bfs-equiv_wf,
equal_wf,
rng_car_wf,
basic-formal-sum_wf,
crng_wf,
formal-sum-mul_wf1,
bfs-equiv-implies,
bfs-reduce_wf,
squash_wf,
true_wf,
rng_sig_wf,
subtype_rel_self,
iff_weakening_equal,
implies-bfs-equiv,
exists_wf,
bag-append_wf,
zero-bfs_wf,
bag_wf,
rng_plus_wf,
bag-map-append,
subtype_rel_bag,
top_wf,
bag-map-map,
bag-map_wf,
rng_times_zero,
rng_times_wf,
crng_times_comm,
crng_times_ac_1,
bfs-equiv-rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
hypothesisEquality,
universeEquality,
because_Cache,
natural_numberEquality,
sqequalRule,
lambdaEquality,
independent_functionElimination,
dependent_functionElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productElimination,
unionElimination,
inlFormation,
productEquality,
inrFormation,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
functionExtensionality,
independent_pairEquality,
independent_pairFormation
Latex:
\mforall{}S:Type. \mforall{}K:CRng. \mforall{}x,x':basic-formal-sum(K;S). \mforall{}k,k':|K|.
bfs-equiv(K;S;x;x') {}\mRightarrow{} bfs-equiv(K;S;k * x;k' * x') supposing k = k'
Date html generated:
2018_05_22-PM-09_45_40
Last ObjectModification:
2018_05_20-PM-10_43_56
Theory : linear!algebra
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