Nuprl Lemma : extend-name-morph-face-map
∀I,K:Cname List. ∀f:name-morph(I;K). ∀z,x:Cname. ∀i:ℕ2.
  (f[z:=x] o (x:=i)) = ((z:=i) o f) ∈ name-morph([z / I];K) supposing (¬(x ∈ K)) ∧ (¬(z ∈ I))
Proof
Definitions occuring in Statement : 
name-comp: (f o g)
, 
face-map: (x:=i)
, 
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
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
and: P ∧ Q
, 
natural_number: $n
, 
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
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
false: False
, 
subtype_rel: A ⊆r B
, 
name-morph: name-morph(I;J)
, 
face-map: (x:=i)
, 
name-comp: (f o g)
, 
extend-name-morph: f[z1:=z2]
, 
compose: f o g
, 
uext: uext(g)
, 
nameset: nameset(L)
, 
coordinate_name: Cname
, 
int_upper: {i...}
, 
nequal: a ≠ b ∈ T 
, 
squash: ↓T
, 
guard: {T}
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
top: Top
, 
le: A ≤ B
, 
less_than: a < b
, 
decidable: Dec(P)
, 
sq_stable: SqStable(P)
, 
rev_implies: P 
⇐ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
isname: isname(z)
, 
true: True
Lemmas referenced : 
name-morphs-equal, 
cons_wf, 
coordinate_name_wf, 
name-comp_wf, 
extend-name-morph_wf, 
face-map_wf, 
l_member_wf, 
istype-void, 
int_seg_wf, 
name-morph_wf, 
nameset_wf, 
eq-cname_wf, 
eq_int_wf, 
int_seg_properties, 
full-omega-unsat, 
intformand_wf, 
intformeq_wf, 
itermVar_wf, 
intformnot_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
int_subtype_base, 
decidable__equal_int, 
istype-le, 
assert-eq-cname, 
bool_wf, 
iff_weakening_uiff, 
assert_wf, 
equal-wf-T-base, 
set_subtype_base, 
le_wf, 
iff_imp_equal_bool, 
le_int_wf, 
btrue_wf, 
iff_functionality_wrt_iff, 
true_wf, 
assert_of_le_int, 
iff_weakening_equal, 
istype-true, 
bfalse_wf, 
false_wf, 
intformle_wf, 
itermConstant_wf, 
intformless_wf, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_less_lemma, 
nsub2_subtype_extd-nameset, 
isname-nameset, 
cons_member, 
isname_wf, 
assert-isname, 
extd-nameset_subtype_int
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
isect_memberFormation_alt, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
introduction, 
extract_by_obid, 
isectElimination, 
hypothesis, 
hypothesisEquality, 
independent_isectElimination, 
sqequalRule, 
productIsType, 
functionIsType, 
universeIsType, 
natural_numberEquality, 
inhabitedIsType, 
because_Cache, 
applyEquality, 
lambdaEquality_alt, 
setElimination, 
rename, 
functionExtensionality, 
applyLambdaEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
independent_functionElimination, 
voidElimination, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
dependent_functionElimination, 
Error :memTop, 
independent_pairFormation, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
equalityIstype, 
promote_hyp, 
instantiate, 
isect_memberEquality_alt, 
cumulativity, 
intEquality, 
dependent_set_memberEquality_alt
Latex:
\mforall{}I,K:Cname  List.  \mforall{}f:name-morph(I;K).  \mforall{}z,x:Cname.  \mforall{}i:\mBbbN{}2.
    (f[z:=x]  o  (x:=i))  =  ((z:=i)  o  f)  supposing  (\mneg{}(x  \mmember{}  K))  \mwedge{}  (\mneg{}(z  \mmember{}  I))
Date html generated:
2020_05_21-AM-10_49_02
Last ObjectModification:
2019_12_08-PM-07_06_29
Theory : cubical!sets
Home
Index