Nuprl Lemma : free-iso-int_wf
∀[S:Type]. ∀[s:S]. free-iso-int(s) ∈ free-vs(ℤ-rng;S) ≅ ℤ supposing ∀x,y:S. (x = y ∈ S)
Proof
Definitions occuring in Statement :
free-iso-int: free-iso-int(s),
free-vs: free-vs(K;S),
vs-iso: A ≅ B,
int-vs: ℤ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
universe: Type,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
all: ∀x:A. B[x],
free-1-iso: free-1-iso(s;K),
vs-iso: A ≅ B,
int_ring: ℤ-rng,
rng_zero: 0,
pi2: snd(t),
pi1: fst(t),
rng_plus: +r,
rng_one: 1,
rng_times: *,
infix_ap: x f y,
exists: ∃x:A. B[x],
free-iso-int: free-iso-int(s),
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
and: P ∧ Q,
vs-map: A ⟶ B,
prop: ℙ,
so_apply: x[s],
top: Top,
free-vs: free-vs(K;S),
vs-point: Point(vs),
mk-vs: mk-vs,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
squash: ↓T,
cand: A c∧ B,
guard: {T},
basic-formal-sum: basic-formal-sum(K;S),
rng_car: |r|,
sq_type: SQType(T),
implies: P ⇒ Q,
one-dim-vs: one-dim-vs(K),
vs-mul: a * x,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
assoc: Assoc(T;op),
comm: Comm(T;op),
record-select: r.x,
record-update: r[x := v],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int-vs: ℤ
Lemmas referenced :
free-1-iso_wf,
int_ring_wf,
istype-universe,
one-dim-int-vs,
pi2_wf,
vs-map_wf,
free-vs_wf,
int-vs_wf,
vs-point_wf,
equal_wf,
pi1_wf_top,
istype-void,
vs-map-eq,
rec_select_update_lemma,
basic-formal-sum_wf,
bfs-equiv_wf,
subtype_base_sq,
int_subtype_base,
bag-summation_wf,
top_wf,
istype-int,
istype-top,
subtype_rel_bag,
subtype_rel_product,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_self,
equal_functionality_wrt_subtype_rel2,
quotient_wf,
bfs-equiv-rel,
mul-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_isectElimination,
axiomEquality,
functionIsType,
because_Cache,
equalityIstype,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
instantiate,
universeEquality,
applyLambdaEquality,
productEquality,
functionEquality,
productElimination,
independent_pairEquality,
voidElimination,
dependent_pairEquality_alt,
productIsType,
functionExtensionality,
dependent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
sqequalBase,
cumulativity,
intEquality,
imageElimination,
independent_pairFormation,
addEquality,
natural_numberEquality,
lambdaFormation_alt,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality
Latex:
\mforall{}[S:Type]. \mforall{}[s:S]. free-iso-int(s) \mmember{} free-vs(\mBbbZ{}-rng;S) \mcong{} \mBbbZ{} supposing \mforall{}x,y:S. (x = y)
Date html generated:
2019_10_31-AM-06_31_09
Last ObjectModification:
2019_08_02-PM-05_30_16
Theory : linear!algebra
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