Nuprl Lemma : qdot-linear
∀[as,bs,cs:ℚ List].
qdot(as;qv-add(bs;cs)) = (qdot(as;bs) + qdot(as;cs)) ∈ ℚ
supposing (dimension(as) = dimension(bs) ∈ ℤ) ∧ (dimension(as) = dimension(cs) ∈ ℤ)
Proof
Definitions occuring in Statement :
qv-add: qv-add(as;bs),
qv-dim: dimension(as),
qdot: qdot(as;bs),
qadd: r + s,
rationals: ℚ,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
qdot: qdot(as;bs),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
qv-dim: dimension(as),
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
rationals_wf,
qsum_wf,
qmul_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
qv-add_wf,
dim-qv-add,
subtype_rel_list,
top_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_wf,
sum_plus_q,
qv-dim_wf,
nat_properties,
subtype_rel_self,
iff_weakening_equal,
select-qv-add,
le_wf,
less_than_wf,
set_subtype_base,
int_subtype_base,
list_wf,
qadd_wf,
qmul_over_plus_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
lambdaEquality_alt,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
universeEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
instantiate,
functionIsType,
dependent_set_memberEquality_alt,
productIsType,
equalityIsType4,
intEquality,
axiomEquality
Latex:
\mforall{}[as,bs,cs:\mBbbQ{} List].
qdot(as;qv-add(bs;cs)) = (qdot(as;bs) + qdot(as;cs))
supposing (dimension(as) = dimension(bs)) \mwedge{} (dimension(as) = dimension(cs))
Date html generated:
2019_10_16-PM-00_34_00
Last ObjectModification:
2018_10_10-AM-11_04_56
Theory : rationals
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