Nuprl Lemma : rng_times_when_r
∀[r:Rng]. ∀[u,v:|r|]. ∀[b:𝔹]. (((when b. u) * v) = (when b. (u * v)) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_when: rng_when,
rng: Rng,
rng_times: *,
rng_car: |r|,
bool: 𝔹,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng_when: rng_when,
mon_when: when b. p,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
rng: Rng,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
bool_wf,
rng_car_wf,
rng_wf,
infix_ap_wf,
rng_times_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_zero,
rng_zero_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
unionElimination,
thin,
equalityElimination,
sqequalRule,
hypothesis,
extract_by_obid,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[u,v:|r|]. \mforall{}[b:\mBbbB{}]. (((when b. u) * v) = (when b. (u * v)))
Date html generated:
2017_10_01-AM-08_19_47
Last ObjectModification:
2017_02_28-PM-02_04_18
Theory : rings_1
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