Nuprl Lemma : face-map-comp-trivial
∀A,B:Cname List. ∀g:name-morph(A;B). ∀x:Cname. ∀i:ℕ2. ((x:=i) o g) = g ∈ name-morph(A;B) supposing ¬(x ∈ A)
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
coordinate_name: Cname,
l_member: (x ∈ l),
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
name-morph: name-morph(I;J),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
name-morph-ext,
name-comp_wf,
nameset_wf,
not_wf,
l_member_wf,
coordinate_name_wf,
int_seg_wf,
name-morph_wf,
list_wf,
face-map_wf,
name-morph_subtype,
cons_wf,
nameset_subtype,
l_subset_right_cons_trivial,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
extd-nameset_wf,
squash_wf,
true_wf,
uext-ap-name,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
equalitySymmetry,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
independent_isectElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
dependent_functionElimination,
sqequalRule,
setElimination,
rename,
unionElimination,
equalityElimination,
equalityTransitivity,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}A,B:Cname List. \mforall{}g:name-morph(A;B). \mforall{}x:Cname. \mforall{}i:\mBbbN{}2. ((x:=i) o g) = g supposing \mneg{}(x \mmember{} A)
Date html generated:
2017_10_05-AM-10_07_38
Last ObjectModification:
2017_07_28-AM-11_16_39
Theory : cubical!sets
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