Nuprl Lemma : bilinear_comm_elim
∀[T:Type]. ∀[pl,tm:T ⟶ T ⟶ T].
(BiLinear(T;pl;tm)) supposing ((∀a,x,y:T. ((a tm (x pl y)) = ((a tm x) pl (a tm y)) ∈ T)) and Comm(T;tm))
Proof
Definitions occuring in Statement :
bilinear: BiLinear(T;pl;tm),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bilinear: BiLinear(T;pl;tm),
comm: Comm(T;op),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
all_wf,
equal_wf,
uall_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesisEquality,
isect_memberEquality,
isectElimination,
because_Cache,
extract_by_obid,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[pl,tm:T {}\mrightarrow{} T {}\mrightarrow{} T].
(BiLinear(T;pl;tm)) supposing
((\mforall{}a,x,y:T. ((a tm (x pl y)) = ((a tm x) pl (a tm y)))) and
Comm(T;tm))
Date html generated:
2017_10_01-AM-08_12_56
Last ObjectModification:
2017_02_28-PM-01_57_11
Theory : gen_algebra_1
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