Nuprl Lemma : extend-name-morph-iota
∀I,K:Cname List. ∀f:name-morph(I;K). ∀z,x:Cname.
(iota(z) o f[z:=x]) = (f o iota(x)) ∈ name-morph(I;[x / K]) supposing (¬(x ∈ K)) ∧ (¬(z ∈ I))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
iota: iota(x),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
coordinate_name: Cname,
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
prop: ℙ,
iota: iota(x),
name-comp: (f o g),
extend-name-morph: f[z1:=z2],
compose: f o g,
uext: uext(g),
ifthenelse: if b then t else f fi ,
btrue: tt,
nameset: nameset(L),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
name-comp_wf,
iota_wf,
extend-name-morph_wf,
name-morph_wf,
not_wf,
l_member_wf,
nameset_wf,
isname-nameset,
eq-cname_wf,
bool_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
set_subtype_base,
le_wf,
int_subtype_base,
isname_wf,
extd-nameset_subtype,
l_subset_right_cons_trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
productEquality,
functionExtensionality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
natural_numberEquality
Latex:
\mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}z,x:Cname.
(iota(z) o f[z:=x]) = (f o iota(x)) supposing (\mneg{}(x \mmember{} K)) \mwedge{} (\mneg{}(z \mmember{} I))
Date html generated:
2017_10_05-AM-10_08_34
Last ObjectModification:
2017_07_28-AM-11_16_55
Theory : cubical!sets
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