Nuprl Lemma : qdot-linear2
∀[as,bs:ℚ List]. ∀[r:ℚ]. qdot(as;qv-mul(r;bs)) = (r * qdot(as;bs)) ∈ ℚ supposing dimension(as) = dimension(bs) ∈ ℤ
Proof
Definitions occuring in Statement :
qv-mul: qv-mul(r;bs),
qv-dim: dimension(as),
qdot: qdot(as;bs),
qmul: r * s,
rationals: ℚ,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qdot: qdot(as;bs),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
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,
qsum_wf,
qmul_wf,
select_wf,
rationals_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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-mul_wf,
dim-qv-mul,
subtype_rel_list,
top_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_wf,
prod_sum_l_q,
qv-dim_wf,
nat_properties,
iff_weakening_equal,
select-qv-mul,
list_wf,
qmul_comm_qrng,
qmul_ac_1_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
sqequalRule,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionEquality,
dependent_set_memberEquality,
axiomEquality
Latex:
\mforall{}[as,bs:\mBbbQ{} List]. \mforall{}[r:\mBbbQ{}].
qdot(as;qv-mul(r;bs)) = (r * qdot(as;bs)) supposing dimension(as) = dimension(bs)
Date html generated:
2018_05_22-AM-00_20_20
Last ObjectModification:
2017_07_26-PM-06_54_55
Theory : rationals
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