Nuprl Lemma : mon_when_hom_swap
∀[g,h:GrpSig]. ∀[f:MonHom(g,h)]. ∀[b:𝔹]. ∀[p:|g|]. ((when b. (f p)) = (f (when b. p)) ∈ |h|)
Proof
Definitions occuring in Statement :
mon_when: when b. p,
monoid_hom: MonHom(M1,M2),
grp_car: |g|,
grp_sig: GrpSig,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mon_when: when b. p,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
monoid_hom: MonHom(M1,M2),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bool_wf,
eqtt_to_assert,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
equal_wf,
grp_car_wf,
monoid_hom_wf,
grp_sig_wf,
squash_wf,
true_wf,
grp_id_wf,
monoid_hom_id,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
baseClosed,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
applyEquality,
setElimination,
rename,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[g,h:GrpSig]. \mforall{}[f:MonHom(g,h)]. \mforall{}[b:\mBbbB{}]. \mforall{}[p:|g|]. ((when b. (f p)) = (f (when b. p)))
Date html generated:
2017_10_01-AM-08_17_17
Last ObjectModification:
2017_02_28-PM-02_02_49
Theory : groups_1
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