Nuprl Lemma : mon_when_thru_op
∀[g:IMonoid]. ∀[b:𝔹]. ∀[p,q:|g|]. ((when b. (p * q)) = ((when b. p) * (when b. q)) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_when: when b. p,
imon: IMonoid,
grp_op: *,
grp_car: |g|,
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,
mon_when: when b. p,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
imon: IMonoid,
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 :
grp_car_wf,
bool_wf,
imon_wf,
infix_ap_wf,
grp_op_wf,
equal_wf,
squash_wf,
true_wf,
grp_id_wf,
mon_ident,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
unionElimination,
thin,
equalityElimination,
sqequalRule,
hypothesis,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[g:IMonoid]. \mforall{}[b:\mBbbB{}]. \mforall{}[p,q:|g|]. ((when b. (p * q)) = ((when b. p) * (when b. q)))
Date html generated:
2017_10_01-AM-08_17_13
Last ObjectModification:
2017_02_28-PM-02_02_09
Theory : groups_1
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